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
Keywords
Examples
a(2) = 4: [(2,0)], [(1,1)], [(1,0),(0,1)], [(0,2)].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..8000 (terms n=101..1000 from Vaclav Kotesovec)
Programs
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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 *)
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