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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)).
2
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
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
Main diagonal gives A293554.
Cf. A007318, A096751 (a similar but different sequence).
Sequence in context: A047030 A047120 A096751 * A099233 A303912 A133815
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Oct 11 2017
STATUS
approved