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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. 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

FORMULA

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

EXAMPLE

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,  ...

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

PROG

(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

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.

Cf. A266941, A297321, A297323, A297325.

Sequence in context: A191588 A106450 A255961 * A055137 A143325 A128888

Adjacent sequences:  A297325 A297326 A297327 * A297329 A297330 A297331

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Dec 28 2017

STATUS

approved

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Last modified March 19 13:22 EDT 2019. Contains 321330 sequences. (Running on oeis4.)