A293551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 141, 86, 22, 1, 1, 1, 9, 36, 105, 216, 331, 310, 160, 30, 1, 1, 1, 10, 45, 148, 357, 672, 855, 692, 282, 42, 1, 1, 1, 11, 55, 201, 554, 1232, 1982, 2214, 1483, 500, 56, 1

Offset

0,9

Comments

A(n,k) is the Euler transform of j -> binomial(j+k-2,k-1) evaluated at n.

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

S. Balakrishnan, S. Govindarajan, and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

N. J. A. Sloane, Transforms

Formula

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

For asymptotics of column k see comment from Vaclav Kotesovec in A255965.

Example

Square array begins:
1,  1,   1,   1,    1,    1,  ...
1,  1,   1,   1,    1,    1,  ...
1,  2,   3,   4,    5,    6,  ...
1,  3,   6,  10,   15,   21,  ...
1,  5,  13,  26,   45,   71,  ...
1,  7,  24,  59,  120,  216,  ...

Maple

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-2, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 17 2017

Mathematica

Table[Function[k, SeriesCoefficient[E^(Sum[x^i/(i (1 - x^i)^k), {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..8 give A000012, A000041, A000219, A000294, A000335, A000391, A000417, A000428, A255965.

Main diagonal gives A293554.

Cf. A007318, A096751 (a similar but different sequence).

Sequence in context: A047030 A047120 A096751 * A099233 A303912 A133815

Adjacent sequences: A293548 A293549 A293550 * A293552 A293553 A293554

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Oct 11 2017

Status

approved