A219555 Number of bipartite partitions of (i,j) with i+j = n into distinct pairs.

1, 2, 4, 10, 19, 38, 73, 134, 242, 430, 749, 1282, 2171, 3622, 5979, 9770, 15802, 25334, 40288, 63560, 99554, 154884, 239397, 367800, 561846, 853584, 1290107, 1940304, 2904447, 4328184, 6422164, 9489940, 13967783, 20480534, 29920277, 43557272, 63194864

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..8000 (terms n=101..1000 from Vaclav Kotesovec)

Formula

a(n) = Sum_{i+j=n} [x^i*y^j] 1/2 * Product_{k,m>=0} (1+x^k*y^m).

G.f.: Product_{k>=1} (1+x^k)^(k+1). - Vaclav Kotesovec, Mar 07 2015

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (1296*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * 3^(4/3) * Zeta(3)^(1/3)) + (3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(5/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 07 2015

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(2 - x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Aug 11 2018

Example

a(2) = 4: [(2,0)], [(1,1)], [(1,0),(0,1)], [(0,2)].

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
       b(n-i*j, min(n-i*j, i-1))*binomial(i+1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Sep 19 2019

Mathematica

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k+1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2015 *)

Crossrefs

Row sums of A054242.

Cf. A026007, A052812, A005380, A255834, A255836.

Sequence in context: A253772 A043330 A295961 * A263738 A011963 A083844

Adjacent sequences: A219552 A219553 A219554 * A219556 A219557 A219558

Keyword

nonn

Author

Alois P. Heinz, Nov 22 2012

Status

approved