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
Examples
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, ...
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
Crossrefs
Programs
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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
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.
Comments