A103650 G.f.: x^2/((1-x^2)^2*Product_{i>0}(1-x^i)).

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

Offset: 1

Author

Vladeta Jovovic, Mar 26 2005

Keywords

• easy
• nonn

Comments

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).

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A014153.

Programs

Mathematica

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 *)

Formula

a(n) = Sum_{k>0} k * A264404(n,k). - Alois P. Heinz, Nov 29 2015

For n>2, a(n) is the Euler transform of [1,3,1,1,1,1,...]. - Benedict W. J. Irwin, Jul 29 2016

a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2). - Vaclav Kotesovec, Jul 30 2016

Extensions

More terms from Robert G. Wilson v, Mar 29 2005