A047654 -id:A047654 - cp's OEIS frontend

A047644 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^19 in powers of x.

1, -19, 171, -969, 3857, -11286, 24206, -34542, 14706, 83011, -294880, 569753, -680694, 220286, 1198672, -3502612, 5661867, -5571579, 791350, 9721976, -23494393, 33415357, -29225230, 2352751, 47086598, -104517176, 140834118, -121255530

Offset: 19

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 19..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 - A047643, A047645 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(19) )); // 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, 19):
seq(a(n), n=19..46);  # Alois P. Heinz, Feb 07 2021

nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^19, {x,0,nmax}], x]//Drop[#, 19] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=19}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^35)); Vec((eta(-x)-1)^19) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=19;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047644_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047644_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^19. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047646 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^21 in powers of x.

Original entry on oeis.org

1, -21, 210, -1330, 5964, -19929, 50253, -91920, 97965, 51604, -526659, 1389297, -2280320, 2118690, 769065, -7613319, 17220042, -23999430, 18024405, 10748850, -63778953, 124134772, -152793270, 99072120, 71722224, -341062407, 610085721

Offset: 21

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 21..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 - A047645, A047647 - A047649, A047654, A047655, A341243.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(21) )); // 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, 21):
seq(a(n), n=21..47);  # Alois P. Heinz, Feb 07 2021

nmax=47; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^21, {x, 0, nmax}], x]//Drop[#, 21] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=21}, 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)^21) \\ Joerg Arndt, Sep 06 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=21;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047646_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047646_list(m); a[k:] # G. C. Greubel, Sep 06 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^21. - G. C. Greubel, Sep 06 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047647 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^22 in powers of x.

Original entry on oeis.org

1, -22, 231, -1540, 7293, -25872, 69971, -140822, 183711, -25102, -634480, 2027804, -3817814, 4439116, -919600, -9829270, 27660479, -44779042, 43632974, -1898820, -92518261, 219961214, -313463842, 267448104, 15757973, -547042056, 1173033400

Offset: 22

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 22..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 - A047646, A047648, A047649, A047654, A047655, A341243.

m:=75;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(22) )); // 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, 22):
seq(a(n), n=22..48);  # Alois P. Heinz, Feb 07 2021

nmax=48; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^22, {x, 0, nmax}], x]//Drop[#, 22] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=22}, 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)^22) \\ Michel Marcus, Sep 05 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=22;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047647_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047647_list(m); a[k:] # G. C. Greubel, Sep 05 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^22. - G. C. Greubel, Sep 05 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A341263 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 - x^k))^n.

Original entry on oeis.org

1, -1, 1, -1, -3, 19, -65, 181, -419, 755, -749, -1530, 12255, -47477, 141065, -343526, 660941, -770917, -911369, 9721976, -40135713, 124134772, -313463842, 631382751, -824406065, -492101356, 8192253811, -35948431288, 115087580857, -299576625051, 627027769120, -894734468883

Offset: 0

Ilya Gutkovskiy, Feb 07 2021

sign

Cf. A001482, A001483, A001484, A001485, A001486, A001487, A001488, A008705, A010815, A047654, A047655, A324595, A340987, A341265.

g:= proc(n) option remember; `if`(n=0, 1, add(add(
     -d, 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, g(n+1),
      (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=0..31);  # Alois P. Heinz, Feb 07 2021

Table[SeriesCoefficient[(-1 + QPochhammer[x, x])^n, {x, 0, 2 n}], {n, 0, 31}]
A[n_, k_] := A[n, k] = If[n == 0, 1, -k Sum[A[n - j, k] DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}];
Table[T[2 n, n], {n, 0, 31}]

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A047644 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^19 in powers of x.

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A047646 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^21 in powers of x.

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A047647 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^22 in powers of x.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A341263 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 - x^k))^n.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica