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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)).

%I A293551 #32 Nov 28 2023 14:27:36
%S A293551 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,
%T A293551 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,
%U A293551 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
%N 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)).
%C A293551 A(n,k) is the Euler transform of j -> binomial(j+k-2,k-1) evaluated at n.
%H A293551 Alois P. Heinz, <a href="/A293551/b293551.txt">Antidiagonals n = 0..140, flattened</a>
%H A293551 S. Balakrishnan, S. Govindarajan, and N. S. Prabhakar, <a href="http://arxiv.org/abs/1105.6231">On the asymptotics of higher-dimensional partitions</a>, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
%H A293551 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A293551 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A293551 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A293551 G.f. of column k: exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).
%F A293551 For asymptotics of column k see comment from _Vaclav Kotesovec_ in A255965.
%e A293551 Square array begins:
%e A293551 1,  1,   1,   1,    1,    1,  ...
%e A293551 1,  1,   1,   1,    1,    1,  ...
%e A293551 1,  2,   3,   4,    5,    6,  ...
%e A293551 1,  3,   6,  10,   15,   21,  ...
%e A293551 1,  5,  13,  26,   45,   71,  ...
%e A293551 1,  7,  24,  59,  120,  216,  ...
%p A293551 with(numtheory):
%p A293551 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
%p A293551       binomial(d+k-2, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
%p A293551     end:
%p A293551 seq(seq(A(n, d-n), n=0..d), d=0..14);  # _Alois P. Heinz_, Oct 17 2017
%t A293551 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
%Y A293551 Columns k=0..8 give A000012, A000041, A000219, A000294, A000335, A000391, A000417, A000428, A255965.
%Y A293551 Main diagonal gives A293554.
%Y A293551 Cf. A007318, A096751 (a similar but different sequence).
%K A293551 nonn,tabl
%O A293551 0,9
%A A293551 _Ilya Gutkovskiy_, Oct 11 2017