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Search: a047654 -id:a047654
Displaying 11-20 of 24 results found.
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A001485
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^7 in powers of x.
(Formerly M4371 N1835)
+10
5
1, -7, 21, -35, 28, 21, -105, 181, -189, 77, 140, -385, 546, -511, 252, 203, -693, 1029, -1092, 798, -203, -581, 1281, -1708, 1687, -1232, 413, 602, -1485, 2233, -2366, 2009, -1099, 14, 1099, -2072, 2667, -2807, 2254, -1477, 0, 1057, -2346, 2744, -3017, 2457
(list; graph; refs; listen; history; text; internal format)
OFFSET
7,2
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 = 7..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 )^7. - 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, 7):
seq(a(n), n=7..52); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax = 52; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^7, {x, 0, nmax}], x] // Drop[#, 7] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^7, {x, 0, 102}], x], 7] (* 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)^7 )); // G. C. Greubel, Sep 04 2023
(SageMath)
m=100; k=7;
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 A001485_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A001485_list(m); a[k:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=70, x='x+O('x^N)); Vec((eta(-x)-1)^7) \\ Joerg Arndt, Sep 04 2023
CROSSREFS
Cf. A001482, A001483, A001484, A001486, A001487, 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
A001486
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^8 in powers of x.
(Formerly M4502 N1906)
+10
5
1, -8, 28, -56, 62, 0, -148, 328, -419, 280, 140, -728, 1232, -1336, 848, 224, -1582, 2688, -3072, 2408, -742, -1568, 3836, -5264, 5306, -3744, 924, 2576, -5686, 7792, -8092, 6272, -2751, -1848, 6008, -9296, 10556, -9800, 6692, -2240, -3206, 8168, -11524
(list; graph; refs; listen; history; text; internal format)
OFFSET
8,2
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 = 8..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 )^8. - 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, 8):
seq(a(n), n=8..50); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax=50; CoefficientList[Series[(Product[(1 -(-x)^j), {j, nmax}] -1)^8, {x, 0, nmax}], x]//Drop[#, 8] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^8, {x, 0, 102}], x], 8] (* 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)^8 )); // G. C. Greubel, Sep 04 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=100; k=8;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001486_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A001486_list(m); a[k:] # G. C. Greubel, Sep 04 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^8) \\ Joerg Arndt, Sep 05 2023
CROSSREFS
Cf. A001482 - A001485, A001487, 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
A047265
Triangle T(n,k), for n >= 1, 1 <= k <= n, read by rows, giving coefficient of x^n in expansion of (Product_{j>=1} (1-(-x)^j) - 1 )^k.
+10
5
1, -1, 1, 0, -2, 1, 0, 1, -3, 1, -1, 0, 3, -4, 1, 0, -2, -1, 6, -5, 1, -1, 2, -3, -4, 10, -6, 1, 0, -2, 6, -3, -10, 15, -7, 1, 0, 2, -6, 12, 0, -20, 21, -8, 1, 0, 1, 6, -16, 19, 9, -35, 28, -9, 1, 0, 0, 0, 16, -35, 24, 28, -56, 36, -10, 1, -1, 2, -3, -6, 40, -65, 21, 62, -84, 45, -11, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
This is an ordinary convolution triangle. If a column k=0 starting at n=0 is added, then this is the Riordan triangle R(1, f(x)), with
f(x) = Product_{j>=1} (1 - (-x)^j) - 1, generating {0, {A121373(n)}_{n>=1}}. - Wolfdieter Lang, Feb 16 2021
LINKS
Alois P. Heinz, Rows n = 1..200, flattened
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
G.f. column k: (Product_{j>=1} (1 - (-x)^j) - 1)^k, for k >= 1. See the name and a Riordan triangle comment above. - Wolfdieter Lang, Feb 16 2021
From G. C. Greubel, Sep 07 2023: (Start)
T(n, n) = 1.
T(n, n-1) = -A000027(n-1).
T(n, n-2) = A000217(n-3).
T(n, n-3) = -A000292(n-5).
Sum_{k=1..n} T(n, k) = (-1)^n * A307059(n).
Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A000041(n). (End)
EXAMPLE
Triangle starts:
1,
-1, 1,
0, -2, 1,
0, 1, -3, 1,
-1, 0, 3, -4, 1,
0, -2, -1, 6, -5, 1,
-1, 2, -3, -4, 10, -6, 1,
0, -2, 6, -3, -10, 15, -7, 1,
0, 2, -6, 12, 0, -20, 21, -8, 1,
0, 1, 6, -16, 19, 9, -35, 28, -9, 1,
0, 0, 0, 16, -35, 24, 28, -56, 36, -10, 1,
-1, 2, -3, -6, 40, ...
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:
T:= proc(n, k) option remember;
`if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
T[n_, k_]:= SeriesCoefficient[(-1)^n*(Product[(1-x^j), {j, n}] - 1)^k, {x, 0, n}];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* Jean-François Alcover, Dec 05 2013 *)
PROG
(PARI) T(n, k) = polcoeff((-1)^n*(Ser(prod(i=1, n, 1-x^i)-1)^k), n) \\ Ralf Stephan, Dec 08 2013
(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
T:= func< n, k | Coefficient(R!( (-1)^n*(-1 + (&*[1 - x^j: j in [1..n]]) )^k ), n) >;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.combinat.q_analogues import q_pochhammer
P.<x> = PowerSeriesRing(ZZ, 50)
def T(n, k): return P( (-1)^n*(-1 + q_pochhammer(n, x, x) )^k ).list()[n]
flatten([[T(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, Sep 07 2023
CROSSREFS
Columns: A001482 - A001488, A001490, A006665, A010815, A047649, A047654, A047655, A047938 - A047648.
Cf. A341418 (differently signed).
Cf. A000027, A000041, A000217, A000292, A121373, A307059.
KEYWORD
sign,easy,nice,tabl
AUTHOR
N. J. A. Sloane
STATUS
approved
A047645
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^20 in powers of x.
+10
4
1, -20, 190, -1140, 4825, -15124, 35320, -57760, 45220, 80560, -405954, 910460, -1289340, 852340, 1259530, -5357924, 10151510, -12048660, 5883350, 12186960, -40135713, 66244280, -69648870, 28191460, 66920755, -195366168, 300881530
(list; graph; refs; listen; history; text; internal format)
OFFSET
20,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 20..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 )^20. - G. C. Greubel, Sep 06 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, 20):
seq(a(n), n=20..46); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] -1)^20, {x, 0, nmax}], x]//Drop[#, 20] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=20}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 06 2023 *)
PROG
(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(20) )); // G. C. Greubel, Sep 06 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=20;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047645_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047645_list(m); a[k:] # G. C. Greubel, Sep 06 2023
(PARI) my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^20) \\ Joerg Arndt, Sep 06 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047638 - A047644, A047646 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047648
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^23 in powers of x.
+10
4
1, -23, 253, -1771, 8832, -33143, 95611, -209231, 317009, -181401, -686642, 2828977, -6099278, 8422623, -4906406, -10919687, 41968146, -78977952, 93297545, -40351223, -117265247, 367581446, -606562624, 631382751, -207879980, -777907725, 2132043121
(list; graph; refs; listen; history; text; internal format)
OFFSET
23,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 23..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 )^23. - 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, 23):
seq(a(n), n=23..49); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax=49; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] -1)^23, {x, 0, nmax}], x]//Drop[#, 23] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=23}, 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)^(23) )); // G. C. Greubel, Sep 05 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=23;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047648_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047648_list(m); a[k:] # G. C. Greubel, Sep 05 2023
(PARI) my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^23) \\ Joerg Arndt, Sep 05 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047638 - A047647, A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047639
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^14 in powers of x.
+10
3
1, -14, 91, -364, 987, -1820, 1897, 754, -8008, 18928, -26845, 19460, 15015, -76272, 141065, -163072, 90727, 99386, -368277, 602616, -643734, 358190, 274547, -1101100, 1801086, -1982330, 1344525, 148316, -2163590, 4032756, -4938843, 4216576
(list; graph; refs; listen; history; text; internal format)
OFFSET
14,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 14..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 )^14. - 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, 14):
seq(a(n), n=14..45); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^14, {x, 0, nmax}], x]//Drop[#, 14] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=14}, 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)^(14) )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=14;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047639_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047639_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^14) \\ Joerg Arndt, Sep 07 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047638, A047640 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047640
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^15 in powers of x.
+10
3
1, -15, 105, -455, 1350, -2793, 3625, -765, -9840, 29120, -48657, 47370, 1680, -111060, 252555, -343526, 267540, 63210, -623510, 1216425, -1495173, 1093210, 166425, -2073645, 3963260, -4864839, 3872295, -618310, -4345470, 9477960, -12611991
(list; graph; refs; listen; history; text; internal format)
OFFSET
15,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 15..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 )^15. - 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, 15):
seq(a(n), n=15..45); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^15, {x, 0, nmax}], x]//Drop[#, 15] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=15}, 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)^(15) )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=15;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047640_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047640_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^15) \\ Joerg Arndt, Sep 07 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047638, A047639, A047641 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047641
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^16 in powers of x.
+10
3
1, -16, 120, -560, 1804, -4128, 6312, -3920, -10530, 42208, -82752, 99584, -39460, -141200, 422568, -673936, 660941, -144720, -938840, 2301568, -3257188, 2916592, -628040, -3492160, 8217536, -11341568, 10408280, -3885040, -7668720, 21033408
(list; graph; refs; listen; history; text; internal format)
OFFSET
16,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 16..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 )^16. - 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, 16):
seq(a(n), n=16..45); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] -1)^16, {x, 0, nmax}], x]//Drop[#, 16] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=16}, 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)^(16) )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=16;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047641_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047641_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^16) \\ Joerg Arndt, Sep 07 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047638 - A047640, A047642 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047642
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^17 in powers of x.
+10
3
1, -17, 136, -680, 2363, -5916, 10319, -9656, -8534, 57426, -133076, 190383, -134810, -140148, 657611, -1240116, 1461337, -770917, -1171504, 4061946, -6678161, 7071269, -3376863, -4939180, 15963612, -25098443, 26265408, -14513461, -10810368, 43792034
(list; graph; refs; listen; history; text; internal format)
OFFSET
17,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 17..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 )^17. - 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, 17):
seq(a(n), n=17..46); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^17, {x, 0, nmax}], x]//Drop[#, 17] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=17}, 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)^(17) )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=17;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047642_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047642_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^17) \\ Joerg Arndt, Sep 07 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047638 - A047641, A047643 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
A047643
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^18 in powers of x.
+10
3
1, -18, 153, -816, 3042, -8262, 16098, -19278, -1377, 72556, -203184, 339030, -326961, -53244, 940050, -2147916, 2975391, -2293488, -911369, 6616332, -12906162, 15883884, -10936899, -4660974, 28758849, -52660134, 62518248, -44501988, -7465464, 84565242
(list; graph; refs; listen; history; text; internal format)
OFFSET
18,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 18..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 )^18. - 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, 18):
seq(a(n), n=18..47); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
nmax=47; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] -1)^18, {x, 0, nmax}], x]//Drop[#, 18] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=18}, 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)^(18) )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=18;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047643_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(k, x) ).list()
a=A047643_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^35)); Vec((eta(-x)-1)^18) \\ Joerg Arndt, Sep 07 2023
CROSSREFS
Cf. A001482 - A001488, A001490, A047638 - A047642, A047644 - A047649, A047654, A047655, A341243.
KEYWORD
sign
AUTHOR
N. J. A. Sloane
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved
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