A286352 -id:A286352 - cp's OEIS frontend

A022602 Expansion of Product_{m>=1} (1+q^m)^(-7).

1, -7, 21, -42, 84, -175, 322, -547, 931, -1561, 2527, -3976, 6167, -9485, 14336, -21280, 31304, -45696, 65940, -94122, 133371, -187734, 262143, -363265, 500381, -685503, 933506, -1263794, 1702590, -2283379, 3047597

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=7 of A286352.

nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

x='x+O('x^50); Vec(prod(m=1, 50, (1 + x^m)^(-7))) \\ Indranil Ghosh, Apr 05 2017

a(n) ~ (-1)^n * 7^(1/4) * exp(Pi*sqrt(7*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(7/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-7*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A022604 Expansion of Product_{m>=1} (1+q^m)^(-9).

Original entry on oeis.org

1, -9, 36, -93, 207, -459, 957, -1827, 3357, -6061, 10620, -18045, 30006, -49122, 79128, -125247, 195435, -301599, 460167, -694026, 1036368, -1534305, 2252277, -3278709, 4736973, -6797196, 9689103, -13722487

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=9 of A286352.

nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^9, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n/2)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(9/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-9*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 3, 3, 4, 0, 1, 0, 1, 4, 6, 4, 5, 0, 1, 0, 2, 5, 9, 10, 5, 6, 0, 1, 0, 2, 8, 13, 16, 15, 6, 7, 0, 1, 0, 2, 9, 21, 26, 25, 21, 7, 8, 0, 1, 0, 2, 12, 27, 44, 45, 36, 28, 8, 9, 0, 1, 0, 3, 15, 40, 63, 80, 71, 49, 36, 9, 10, 0, 1

Offset: 0

Ilya Gutkovskiy, Feb 08 2021

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  0,  1;
  0,  1,  0,  1;
  0,  1,  2,  0,  1;
  0,  1,  2,  3,  0,  1;
  0,  1,  3,  3,  4,  0,  1;
  0,  1,  4,  6,  4,  5,  0,  1;
  0,  2,  5,  9, 10,  5,  6,  0,  1;
  0,  2,  8, 13, 16, 15,  6,  7,  0,  1;
  0,  2,  9, 21, 26, 25, 21,  7,  8,  0,  1;
  0,  2, 12, 27, 44, 45, 36, 28,  8,  9,  0,  1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give A000007, A000700, A338463, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253.
Main diagonal and lower diagonals give A000012, A000004, A001477, A000217, A000290.
Row sums give A307058.
T(2n,n) gives A341265.
Cf. A000009, A047265, A060642, A286352, A308680.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -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=0..n), n=0..12);  # Alois P. Heinz, Feb 09 2021

T[n_, k_] := SeriesCoefficient[(-1 + 2/QPochhammer[-1, -x])^k, {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

G.f. of column k: (-1 + Product_{j>=1} (1 + x^(2*j-1)))^k.

Sum_{k=0..n} (-1)^(n-k) * T(n,k) = A000009(n).

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A022602 Expansion of Product_{m>=1} (1+q^m)^(-7).

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A022604 Expansion of Product_{m>=1} (1+q^m)^(-9).

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A341279 Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

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