A001490 -id:A001490 - cp's OEIS frontend

A001488 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^10 in powers of x.

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

Offset: 10

N. J. A. Sloane

sign

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).

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)

Cf. A001482 - A001487, A001490, A047638 - A047649, A047654, A047655, A341243.

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^10 )); // G. C. Greubel, Sep 04 2023

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

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 *)

my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^10) \\ Joerg Arndt, Sep 05 2023

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. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001488_list(m); a[k:] # G. C. Greubel, Sep 04 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^10. - G. C. Greubel, Sep 04 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047654 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.

Original entry on oeis.org

1, -2, 1, 0, -2, 2, -2, 2, 1, 0, 2, -2, 3, 0, 2, 0, 0, 2, -2, 0, -2, 2, -1, 0, 0, -2, -2, -2, 1, -2, 0, -2, -2, 0, 2, 0, -2, 0, -2, 0, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, -2, 2, 2, 0, 2, 0, 2, 0, 2, 2, 0, -4, 0, 0, 2, 1, -2, 0, -2, 0, 0, 0, 0, 2, -4, 1, 0, 0, -2, -2, -2, -2, 0, 0, -2, 0, 2, -2, 2, -2

Offset: 2

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 2..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.

Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047655, A341243.

m:=120;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^2 )); // G. C. Greubel, Sep 07 2023

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, 2):
seq(a(n), n=2..94);  # Alois P. Heinz, Feb 07 2021

nmax=94; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^2, {x, 0, nmax}], x]//Drop[#, 2] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=2}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

seq(n)={Vec((prod(j=1, n, 1-(-x)^j + O(x^n)) - 1)^2)} \\ Andrew Howroyd, Feb 07 2021

from sage.modular.etaproducts import qexp_eta
m=125; k=2;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047654_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047654_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^2. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047655 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^3 in powers of x.

Original entry on oeis.org

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

Offset: 3

N. J. A. Sloane

sign

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.

Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047654, A341243.

m:=120;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^3 )); // G. C. Greubel, Sep 07 2023

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

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 *)

my(x='x+O('x^99)); Vec((eta(-x)-1)^3) \\ Joerg Arndt, Sep 07 2023

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. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047655_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^3. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047649 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^11 in powers of x.

Original entry on oeis.org

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

Offset: 11

N. J. A. Sloane

sign

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.

Cf. A001482 - A001488, A001490, A047638 - A047648, A047654, A047655, A341243.

m:=75;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(11) )); // G. C. Greubel, Sep 05 2023

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

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 *)

my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^11) \\ Joerg Arndt, Sep 05 2023

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. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047649_list(m); a[k:] # G. C. Greubel, Sep 05 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^11. - G. C. Greubel, Sep 05 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047638 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^13 in powers of x.

Original entry on oeis.org

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

Offset: 13

N. J. A. Sloane

sign

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)

Cf. A001482 - A001488, A001490, A047639 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(13) )); // G. C. Greubel, Sep 07 2023

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

With[{k=13}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^40)); Vec((eta(-x)-1)^13) \\ Joerg Arndt, Sep 07 2023

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. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047638_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^13. - G. C. Greubel, Sep 07 2023

Definition corrected by Robert Israel, Aug 08 2018

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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

			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, ...

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)

Columns: A001482 - A001488, A001490, A006665, A010815, A047649, A047654, A047655, A047938 - A047648.
Cf. A341418 (differently signed).
Cf. A000027, A000041, A000217, A000292, A121373, A307059.

R:=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

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

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 *)

T(n,k) = polcoeff((-1)^n*(Ser(prod(i=1,n,1-x^i)-1)^k), n) \\ Ralf Stephan, Dec 08 2013

from sage.combinat.q_analogues import q_pochhammer
P. = 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

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)

A047645 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^20 in powers of x.

Original entry on oeis.org

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

Offset: 20

N. J. A. Sloane

sign

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.

Cf. A001482 - A001488, A001490, A047638 - A047644, A047646 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(20) )); // G. C. Greubel, Sep 06 2023

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

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 *)

my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^20) \\ Joerg Arndt, Sep 06 2023

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. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047645_list(m); a[k:] # G. C. Greubel, Sep 06 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^20. - G. C. Greubel, Sep 06 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047648 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^23 in powers of x.

Original entry on oeis.org

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

Offset: 23

N. J. A. Sloane

sign

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.

Cf. A001482 - A001488, A001490, A047638 - A047647, A047649, A047654, A047655, A341243.

m:=75;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(23) )); // G. C. Greubel, Sep 05 2023

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

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 *)

my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^23) \\ Joerg Arndt, Sep 05 2023

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. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047648_list(m); a[k:] # G. C. Greubel, Sep 05 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^23. - G. C. Greubel, Sep 05 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047639 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^14 in powers of x.

Original entry on oeis.org

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

Offset: 14

N. J. A. Sloane

sign

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)

Cf. A001482 - A001488, A001490, A047638, A047640 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(14) )); // G. C. Greubel, Sep 07 2023

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

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 *)

my(x='x+O('x^40)); Vec((eta(-x)-1)^14) \\ Joerg Arndt, Sep 07 2023

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. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047639_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^14. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047640 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^15 in powers of x.

Original entry on oeis.org

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

Offset: 15

N. J. A. Sloane

sign

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)

Cf. A001482 - A001488, A001490, A047638, A047639, A047641 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(15) )); // G. C. Greubel, Sep 07 2023

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

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 *)

my(x='x+O('x^40)); Vec((eta(-x)-1)^15) \\ Joerg Arndt, Sep 07 2023

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. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047640_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^15. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

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