A047648 -id:A047648 - cp's OEIS frontend

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

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

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)

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

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

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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.

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

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Magma

Maple

Mathematica

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