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A002763
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Number of bipartite partitions.
(Formerly M3414 N1380)
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3
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4, 11, 26, 52, 98, 171, 289, 467, 737, 1131, 1704, 2515, 3661, 5246, 7430, 10396, 14405, 19760, 26884, 36269, 48583, 64614, 85399, 112170, 146526, 190362, 246099, 316621, 405556, 517224, 657012, 831320, 1048055, 1316611, 1648486, 2057324, 2559719, 3175309
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OFFSET
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0,1
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REFERENCES
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M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 11.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: (4 - x - 3*x^2 + x^3) / ((1-x)^3 * (1+x)) * Product_{k>=1} 1/(1-x^k).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 3*sqrt(n)/(2*sqrt(2)*Pi^3).
(End)
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MAPLE
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with(numtheory):
b:= proc(n, k) option remember;
`if`(n>k, 0, 1) +`if`(isprime(n), 0,
add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
end:
a:= n-> b((45*2^n)$2):
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MATHEMATICA
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b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[45*2^n, 45*2^n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[(4 - x - 3*x^2 + x^3) / ((1 - x)^3 * (1 + x)) / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2017 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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