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A219555
Number of bipartite partitions of (i,j) with i+j = n into distinct pairs.
15
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.
Sequence in context: A253772 A043330 A295961 * A263738 A011963 A083844
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 22 2012
STATUS
approved