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A103650
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G.f.: x^2/((1-x^2)^2*Product_{i>0}(1-x^i)).
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3
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0, 1, 1, 4, 5, 12, 16, 31, 42, 72, 98, 155, 210, 315, 423, 610, 812, 1136, 1498, 2047, 2674, 3585, 4642, 6125, 7865, 10240, 13046, 16791, 21237, 27060, 33993, 42933, 53591, 67155, 83332, 103687, 127956, 158196, 194217, 238720, 291663, 356582
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OFFSET
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1,4
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COMMENTS
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Let pi be a partition of n and b(pi,k) = Sum p, where p runs over all distinct parts p of pi whose multiplicities are >=k. Let T(n,k) = Sum b(pi,k), when pi runs over all partitions pi of n. G.f. for T(n,k) is x^k/((1-x^k)^2*Product_{i>0}(1-x^i)). a(n) = T(n,2).
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LINKS
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FORMULA
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a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2). - Vaclav Kotesovec, Jul 30 2016
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EXAMPLE
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Partitions of 4 are [1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4] and a(4) = 1 + 1 + 2 + 0 + 0 = 4.
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MATHEMATICA
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Drop[ CoefficientList[ Series[ x^2/((1 - x^2)^2*Product[(1 - x^i), {i, 50}]), {x, 0, 42}], x], 1] (* Robert G. Wilson v, Mar 29 2005 *)
Table[Sum[PartitionsP[k]*(n-k)*(1 + (-1)^(n-k))/4, {k, 0, n}], {n, 1, 50}] (* Vaclav Kotesovec, Jul 30 2016 *)
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CROSSREFS
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KEYWORD
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easy,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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