A022662 -id:A022662 - cp's OEIS frontend

A297323 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^k.

1, 1, 0, 1, -1, 0, 1, -2, -2, 0, 1, -3, -3, -1, 0, 1, -4, -3, 2, -1, 0, 1, -5, -2, 8, 4, 5, 0, 1, -6, 0, 16, 9, 16, 1, 0, 1, -7, 3, 25, 9, 18, -3, 13, 0, 1, -8, 7, 34, 0, 4, -35, 6, 4, 0, 1, -9, 12, 42, -21, -26, -90, -33, -31, 0, 0, 1, -10, 18, 48, -56, -66, -145, -56, -66, -72, 2, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			G.f. of column k: A_k(x) = 1 - k*x + (1/2)*k*(k - 5)*x^2 - (1/6)*k*(k^2 - 15*k + 20)*x^3 + (1/24)*k*(k^3 - 30*k^2 + 155*k - 150)*x^4 - (1/120)*k*(k^4 - 50*k^3 + 575*k^2 - 1750*k + 624)*x^5 + ...
Square array begins:
  1,  1,   1,   1,   1,   1,  ...
  0, -1,  -2,  -3,  -4,  -5,  ...
  0, -2,  -3,  -3,  -2,   0,  ...
  0, -1,   2,   8,  16,  25,  ...
  0, -1,   4,   9,   9,   0,  ...
  0,  5,  16,  18,   4, -26,  ...

Columns k=0..32 give A000007, A022661, A022662, A022663, A022664, A022665, A022666, A022667, A022668, A022669, A022670, A022671, A022672, A022673, A022674, A022675, A022676, A022677, A022678, A022679, A022680, A022681, A022682, A022683, A022684, A022685, A022686, A022687, A022688, A022689, A022690, A022691, A022692.
Main diagonal gives A297324.
Antidiagonal sums give A299209.
Cf. A266964, A297321, A297325, A297328.

Table[Function[k, SeriesCoefficient[Product[(1 - i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, (1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ Iain Fox, Dec 28 2017

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

A022726 Expansion of 1/Product_{m>=1} (1 - m*q^m)^2.

Original entry on oeis.org

1, 2, 7, 18, 49, 114, 282, 624, 1422, 3058, 6597, 13700, 28564, 57698, 116479, 230398, 453698, 879080, 1696732, 3230578, 6124326, 11486884, 21439480, 39659598, 73036175, 133445640, 242756058, 438680734, 789328034, 1411926186, 2515574329, 4458203590, 7871211452, 13831782146

Offset: 0

N. J. A. Sloane

nonn

Self-convolution of A006906. - Vaclav Kotesovec, Jan 06 2016

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000

Cf. A006906, A022662.

Column k=2 of A297328.

n:=40; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^2:m in [1..n]])); // G. C. Greubel, Jul 25 2018

nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 06 2016 *)

N=66; q='q+O('q^N);
gf= 1/prod(n=1,N, (1-n*q^n)^2 );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

From Vaclav Kotesovec, Jan 07 2016: (Start)
a(n) ~ c * n * 3^(n/3), where
c = 9588921272.54120308291761424720457... = (c0^2 + 2*c1*c2)/3  if mod(n,3)=0
c = 9588921272.50566179874517327053929... = (c2^2 + 2*c0*c1)/3  if mod(n,3)=1
c = 9588921272.49785814355801212400055... = (c1^2 + 2*c0*c2)/3  if mod(n,3)=2
For the constants c0, c1, c2 see A006906.
(End)
G.f.: exp(2*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

Added more terms, Joerg Arndt, Oct 06 2012

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A297323 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^k.

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A022726 Expansion of 1/Product_{m>=1} (1 - m*q^m)^2.

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