A297328 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^k.
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 14, 0, 1, 5, 18, 37, 49, 25, 0, 1, 6, 25, 64, 114, 114, 56, 0, 1, 7, 33, 100, 219, 312, 282, 97, 0, 1, 8, 42, 146, 375, 676, 855, 624, 198, 0, 1, 9, 52, 203, 594, 1276, 2030, 2178, 1422, 354, 0, 1, 10, 63, 272, 889, 2196, 4155, 5736, 5496, 3058, 672, 0
Offset: 0
Examples
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, 3, 7, 12, 18, 25, ... 0, 6, 18, 37, 64, 100, ... 0, 14, 49, 114, 219, 375, ... 0, 25, 114, 312, 676, 1276, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..200, flattened
Crossrefs
Columns k=0..32 give A000007, A006906, A022726, A022727, A022728, A022729, A022730, A022731, A022732, A022733, A022734, A022735, A022736, A022737, A022738, A022739, A022740, A022741, A022742, A022743, A022744, A022745, A022746, A022747, A022748, A022749, A022750, A022751, A022752, A022753, A022754, A022755, A022756.
Main diagonal gives A297329.
Antidiagonal sums give A299162.
Programs
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Mathematica
Table[Function[k, SeriesCoefficient[Product[1/(1 - i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten -
PARI
first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, 1/(1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ Iain Fox, Dec 28 2017
Formula
G.f. of column k: Product_{j>=1} 1/(1 - j*x^j)^k.
A(0,k) = 1; A(n,k) = (k/n) * Sum_{j=1..n} A078308(j) * A(n-j,k). - Seiichi Manyama, Aug 16 2023