%I #30 Sep 19 2019 20:48:10

%S 1,2,4,10,19,38,73,134,242,430,749,1282,2171,3622,5979,9770,15802,

%T 25334,40288,63560,99554,154884,239397,367800,561846,853584,1290107,

%U 1940304,2904447,4328184,6422164,9489940,13967783,20480534,29920277,43557272,63194864

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

%H Alois P. Heinz, <a href="/A219555/b219555.txt">Table of n, a(n) for n = 0..8000</a> (terms n=101..1000 from Vaclav Kotesovec)

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

%F G.f.: Product_{k>=1} (1+x^k)^(k+1). - _Vaclav Kotesovec_, Mar 07 2015

%F 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

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

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

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

%p b(n-i*j, min(n-i*j, i-1))*binomial(i+1, j), j=0..n/i)))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..42); # _Alois P. Heinz_, Sep 19 2019

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

%Y Row sums of A054242.

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

%K nonn

%O 0,2

%A _Alois P. Heinz_, Nov 22 2012