Search: a047654 -id:a047654
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A001482
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^4 in powers of x.
(Formerly M3263 N1317)
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+10
27
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1, -4, 6, -4, -3, 12, -16, 16, -6, -8, 18, -28, 26, -20, 2, 12, -23, 32, -36, 28, -6, 4, 22, -20, 39, -32, 32, -12, 2, 16, -12, 24, -40, 28, -34, 0, -6, -16, 0, -40, 6, -36, 26, -32, -5, 0, -20, 8, -16, 12, -10, 40, -22, 12, 14, 12, 45, 16, 38, 4, 12, 0, 34, 8, 38, 12, -24, 44, 2, 16
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OFFSET
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4,2
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REFERENCES
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H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 4):
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MATHEMATICA
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nmax = 73; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(1 -QPochhammer[-x])^4, {x, 0, 100}], x], 4] (* G. C. Greubel, Sep 04 2023 *)
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PROG
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(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (1 - (&*[1-(-x)^j: j in [1..m+2]]))^4 )); // G. C. Greubel, Sep 04 2023
(SageMath)
m=100
def f4(x): return (1 - product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1, m+2) ) )^4
P.<x> = PowerSeriesRing(QQ, prec)
return P( f4(x) ).list()
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^4) \\ Joerg Arndt, Sep 04 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A001488
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^10 in powers of x.
(Formerly M4703 N2010)
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+10
25
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1, -10, 45, -120, 200, -162, -160, 810, -1530, 1730, -749, -1630, 4755, -7070, 6700, -2450, -5295, 14070, -20010, 19350, -10157, -6290, 25515, -40660, 44940, -34268, 9180, 24510, -57195, 78060, -79087, 56610, -13935, -39600, 89805, -121638, 125405
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graph;
refs;
listen;
history;
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internal format)
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OFFSET
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10,2
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^10. - G. C. Greubel, Sep 04 2023
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 10):
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MATHEMATICA
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nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] -1)^10, {x, 0, nmax}], x]//Drop[#, 10] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^10, {x, 0, 102}], x], 10] (* G. C. Greubel, Sep 04 2023 *)
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PROG
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(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^10 )); // G. C. Greubel, Sep 04 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=100; k=10;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^10) \\ Joerg Arndt, Sep 05 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A047655
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^3 in powers of x.
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+10
25
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1, -3, 3, -1, -3, 6, -6, 6, 0, -3, 6, -9, 8, -6, 0, 0, -6, 6, -13, 3, -6, 3, 0, -3, 6, -9, 6, -3, 6, 0, 6, 6, -3, 11, 0, 6, 0, 9, 0, 0, 0, -3, 13, 0, 0, -6, 0, -6, 3, -3, -6, 0, -15, -6, -3, 0, -6, 0, -6, 0, -6, -6, 0, -11, 0, 0, -6, 0, 6, 0, 6, 0, 0, 0, -3, 19, 12, -3, 0, 0, 6, 6, 6, 6, 0, 0, 6, 0, 21, 3
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OFFSET
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3,2
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^3. - G. C. Greubel, Sep 07 2023
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 3):
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MATHEMATICA
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nmax=92; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^3, {x, 0, nmax}], x]//Drop[#, 3] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=3}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)
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PROG
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(Magma)
m:=120;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^3 )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=125; k=3;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
(PARI) my(x='x+O('x^99)); Vec((eta(-x)-1)^3) \\ Joerg Arndt, Sep 07 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A047649
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^11 in powers of x.
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+10
23
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1, -11, 55, -165, 319, -352, -44, 1100, -2585, 3542, -2519, -1530, 8085, -14410, 16170, -9460, -6644, 28105, -46145, 50248, -32802, -6193, 57200, -102575, 121968, -100397, 35123, 60390, -158840, 226413, -234344, 168773, -37070, -131175, 290851, -391402
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OFFSET
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11,2
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^11. - G. C. Greubel, Sep 05 2023
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 11):
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MATHEMATICA
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nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^11, {x, 0, nmax}], x]//Drop[#, 11] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=11}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 05 2023 *)
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PROG
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(Magma)
m:=75;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(11) )); // G. C. Greubel, Sep 05 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=11;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^11) \\ Joerg Arndt, Sep 05 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A047638
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^13 in powers of x.
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+10
22
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1, -13, 78, -286, 702, -1131, 845, 1300, -5928, 11583, -13715, 5915, 15834, -47477, 73658, -71201, 20436, 79391, -198796, 280345, -258557, 92807, 200850, -536341, 773916, -768222, 432705, 204477, -979628, 1626196, -1856569, 1471184, -452192
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OFFSET
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13,2
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^13. - G. C. Greubel, Sep 07 2023
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MAPLE
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N:= 100: # to get a(13)..a(N)
G:= (mul(1-(-x)^j, j=1..N)-1)^13:
S:= series(G, x, N+1):
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MATHEMATICA
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With[{k=13}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)
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PROG
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(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(13) )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=13;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
(PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^13) \\ Joerg Arndt, Sep 07 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A001490
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Expansion of {Product_{j>=1} (1 - (-x)^j) - 1}^12 in powers of x.
(Formerly M4845 N2071)
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+10
17
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1, -12, 66, -220, 483, -660, 252, 1320, -4059, 6644, -6336, 240, 12255, -27192, 35850, -27972, -2343, 50568, -99286, 122496, -96162, 11584, 115116, -242616, 315216, -283800, 128304, 126280, -409398, 622644, -671550, 501468, -122508, -382360
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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REFERENCES
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H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: (eta(z)*eta(6*z)/(eta(2*z)*eta(3*z)))^12.
a(n) = [x^n]( QPochhammer(-x) - 1 )^12. - G. C. Greubel, Sep 05 2023
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MATHEMATICA
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With[{k=12}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 102}], x], k]] (* G. C. Greubel, Sep 04 2023 *)
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PROG
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(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(12) )); // G. C. Greubel, Sep 05 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=100; k=12;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^12) \\ Joerg Arndt, Sep 05 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A338463
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Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2.
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+10
11
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1, 0, 2, 2, 3, 4, 5, 8, 9, 12, 15, 20, 23, 28, 36, 44, 52, 62, 76, 90, 106, 124, 149, 176, 203, 236, 279, 324, 372, 430, 499, 576, 657, 752, 867, 992, 1124, 1280, 1463, 1662, 1876, 2124, 2410, 2722, 3061, 3446, 3889, 4374, 4896, 5490, 6166, 6900, 7700, 8600
(list;
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internal format)
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OFFSET
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2,3
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LINKS
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FORMULA
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G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^2.
a(n) = [x^n]( (2/QPochhammer(-1,-x) - 1)^2 ). - G. C. Greubel, Sep 07 2023
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MATHEMATICA
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nmax = 55; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
With[{k=2}, Drop[CoefficientList[Series[(2/QPochhammer[-1, -x] -1)^k, {x, 0, 80}], x], k]] (* G. C. Greubel, Sep 07 2023 *)
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PROG
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(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (-1 + (&*[1+x^(2*j+1): j in [0..m+2]]) )^2 )); // G. C. Greubel, Sep 07 2023
(SageMath)
m=80
def f(x): return (-1 + product(1+x^(2*j-1) for j in range(1, m+3)) )^2
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(x) ).list()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A001483
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^5 in powers of x.
(Formerly M3791 N1546)
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+10
6
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1, -5, 10, -10, 0, 19, -35, 40, -25, -10, 45, -75, 80, -60, 15, 45, -85, 115, -115, 90, -21, -35, 95, -130, 135, -135, 70, -35, -65, 105, -146, 120, -150, 90, -65, -25, 90, -115, 150, -125, 130, -45, 80, 35, -5, 160, -110, 170, -85, 95, 25, 50, 0, -60, 95, -116, 120, -135
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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5,2
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^5. - G. C. Greubel, Sep 04 2023
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 5):
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MATHEMATICA
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nmax = 62; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^5, {x, 0, nmax}], x] // Drop[#, 5] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^5, {x, 0, 102}], x], 5] (* G. C. Greubel, Sep 04 2023 *)
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PROG
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(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^5 )); // G. C. Greubel, Sep 04 2023
(SageMath)
m=100; k=5;
def f(k, x): return (-1 + product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1, m+2) ) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^5) \\ Joerg Arndt, Sep 04 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A001487
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^9 in powers of x.
(Formerly M4618 N1971)
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+10
6
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1, -9, 36, -84, 117, -54, -177, 540, -837, 755, -54, -1197, 2535, -3204, 2520, -246, -3150, 6426, -8106, 7011, -2844, -3549, 10359, -15120, 15804, -11403, 2574, 8610, -18972, 25425, -25824, 18954, -6165, -10080, 25101, -35262, 37799, -31374, 17379, 1929
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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9,2
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = [x^n]( QPochhammer(-x) - 1 )^9. - G. C. Greubel, Sep 04 2023
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n, 9):
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MATHEMATICA
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nmax=48; CoefficientList[Series[(Product[(1 - (-x)^j), {j, nmax}] -1)^9, {x, 0, nmax}], x]//Drop[#, 9] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^9, {x, 0, 102}], x], 9] (* G. C. Greubel, Sep 04 2023 *)
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PROG
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(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^9 )); // G. C. Greubel, Sep 04 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=100; k=9;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^9) \\ Joerg Arndt, Sep 05 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A001484
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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^6 in powers of x.
(Formerly M4107 N1704)
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+10
5
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1, -6, 15, -20, 9, 24, -65, 90, -75, 6, 90, -180, 220, -180, 66, 110, -264, 360, -365, 264, -66, -178, 375, -510, 496, -414, 180, 60, -330, 570, -622, 582, -390, 220, 96, -300, 621, -630, 705, -492, 300, 0, -235, 420, -570, 594, -735, 420, -420, -120, 219, -586, 360
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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6,2
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = [x^n] ( QPochhammer(-x) - 1 )^6. - G. C. Greubel, Sep 04 2023
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MAPLE
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N:= 100:
S:= series((mul(1-(-x)^j, j=1..N)-1)^6, x, N+1):
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MATHEMATICA
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Drop[CoefficientList[Series[(QPochhammer[-x] -1)^6, {x, 0, 102}], x], 6] (* G. C. Greubel, Sep 04 2023 *)
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PROG
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(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^6 )); // G. C. Greubel, Sep 04 2023
(SageMath)
m=100; k=6;
def f(k, x): return (-1 + product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1, m+2) ) )^k
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^6) \\ Joerg Arndt, Sep 04 2023
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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