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Search: a047654 -id:a047654
Displaying 1-10 of 24 results found. page 1 2 3
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A001482 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^4 in powers of x.
(Formerly M3263 N1317) 		+10
27
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
4,2
REFERENCES
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).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 4..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
MAPLE
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):
seq(a(n), n=4..73); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
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 *)
PROG
(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
def A001482_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f4(x) ).list()
a=A001482_list(m); a[4:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^4) \\ Joerg Arndt, Sep 04 2023
CROSSREFS
Cf. A001483 - A001488, A047638 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A001488 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^10 in powers of x.
(Formerly M4703 N2010) 		+10
25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
10,2
REFERENCES
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).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 10..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
FORMULA
a(n) = [x^n]( QPochhammer(-x) - 1 )^10. - G. C. Greubel, Sep 04 2023
MAPLE
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):
seq(a(n), n=10..46); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
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 *)
PROG
(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
def A001488_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A001488_list(m); a[k:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^10) \\ Joerg Arndt, Sep 05 2023
CROSSREFS
Cf. A001482 - A001487, A001490, A047638 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047655 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^3 in powers of x. +10
25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 3..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
FORMULA
a(n) = [x^n]( QPochhammer(-x) - 1 )^3. - G. C. Greubel, Sep 07 2023
MAPLE
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):
seq(a(n), n=3..92); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
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 *)
PROG
(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
def A047655_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047655_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^99)); Vec((eta(-x)-1)^3) \\ Joerg Arndt, Sep 07 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047654, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047649 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^11 in powers of x. +10
23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
11,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 11..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
FORMULA
a(n) = [x^n]( QPochhammer(-x) - 1 )^11. - G. C. Greubel, Sep 05 2023
MAPLE
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):
seq(a(n), n=11..46); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
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 *)
PROG
(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
def A047649_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047649_list(m); a[k:] # G. C. Greubel, Sep 05 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^11) \\ Joerg Arndt, Sep 05 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047638 - A047648, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047638 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^13 in powers of x. +10
22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
13,2
LINKS
Robert Israel, Table of n, a(n) for n = 13..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
FORMULA
a(n) = [x^n]( QPochhammer(-x) - 1 )^13. - G. C. Greubel, Sep 07 2023
MAPLE
N:= 100: # to get a(13)..a(N)
G:= (mul(1-(-x)^j, j=1..N)-1)^13:
S:= series(G, x, N+1):
seq(coeff(S, x, n), n=13..N); # Robert Israel, Aug 08 2018
MATHEMATICA
With[{k=13}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)
PROG
(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
def A047638_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047638_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^13) \\ Joerg Arndt, Sep 07 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047639 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition corrected by Robert Israel, Aug 08 2018
STATUS
approved
A001490 Expansion of {Product_{j>=1} (1 - (-x)^j) - 1}^12 in powers of x.
(Formerly M4845 N2071) 		+10
17
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)
OFFSET
1,2
REFERENCES
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).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
FORMULA
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
MATHEMATICA
With[{k=12}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 102}], x], k]] (* G. C. Greubel, Sep 04 2023 *)
PROG
(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
def A001490_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A001490_list(m); a[k:] # G. C. Greubel, Sep 05 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^12) \\ Joerg Arndt, Sep 05 2023
CROSSREFS
Cf. A001482 - A001488, A047638 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
STATUS
approved
A338463 Expansion of g.f.: (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^2. +10
11
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; graph; refs; listen; history; text; internal format)
OFFSET
2,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 2..1002
FORMULA
G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^2.
a(n) = Sum_{k=1..n-1} A000700(k) * A000700(n-k).
a(n) = A073252(n) - 2 * A000700(n) for n > 0.
a(n) = [x^n]( (2/QPochhammer(-1,-x) - 1)^2 ). - G. C. Greubel, Sep 07 2023
MATHEMATICA
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 *)
PROG
(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
def A338463_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(x) ).list()
a=A338463_list(m); a[2:] # G. C. Greubel, Sep 07 2023
CROSSREFS
Cf. A000700, A022597, A047654, A048574, A073252, A327380, A338223.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 31 2021
STATUS
approved
A001483 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^5 in powers of x.
(Formerly M3791 N1546) 		+10
6
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)
OFFSET
5,2
REFERENCES
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).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 5..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
FORMULA
a(n) = [x^n]( QPochhammer(-x) - 1 )^5. - G. C. Greubel, Sep 04 2023
MAPLE
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):
seq(a(n), n=5..62); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
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 *)
PROG
(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
def A001483_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A001483_list(m); a[k:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^5) \\ Joerg Arndt, Sep 04 2023
CROSSREFS
Cf. A001482, A001484 - A001488, A047638 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A001487 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^9 in powers of x.
(Formerly M4618 N1971) 		+10
6
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)
OFFSET
9,2
REFERENCES
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).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 9..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
FORMULA
a(n) = [x^n]( QPochhammer(-x) - 1 )^9. - G. C. Greubel, Sep 04 2023
MAPLE
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):
seq(a(n), n=9..48); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
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 *)
PROG
(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
def A001487_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A001487_list(m); a[k:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^9) \\ Joerg Arndt, Sep 05 2023
CROSSREFS
Cf. A001482 - A001486, A001488, A047638 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A001484 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^6 in powers of x.
(Formerly M4107 N1704) 		+10
5
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)
OFFSET
6,2
REFERENCES
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).
LINKS
Robert Israel, Table of n, a(n) for n = 6..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
FORMULA
a(n) = [x^n] ( QPochhammer(-x) - 1 )^6. - G. C. Greubel, Sep 04 2023
MAPLE
N:= 100:
S:= series((mul(1-(-x)^j, j=1..N)-1)^6, x, N+1):
seq(coeff(S, x, j), j=6..N); # Robert Israel, Feb 05 2019
MATHEMATICA
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^6, {x, 0, 102}], x], 6] (* G. C. Greubel, Sep 04 2023 *)
PROG
(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
def A001484_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A001484_list(m); a[k:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^6) \\ Joerg Arndt, Sep 04 2023
CROSSREFS
Cf. A001482, A001483, A001485 - A001488, A047638 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Edited by Robert Israel, Feb 05 2019
STATUS
approved
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Last modified August 27 19:37 EDT 2024. Contains 375471 sequences. (Running on oeis4.)