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%I #17 Aug 02 2016 04:29:46
%S 0,1,1,4,5,12,16,31,42,72,98,155,210,315,423,610,812,1136,1498,2047,
%T 2674,3585,4642,6125,7865,10240,13046,16791,21237,27060,33993,42933,
%U 53591,67155,83332,103687,127956,158196,194217,238720,291663,356582
%N G.f.: x^2/((1-x^2)^2*Product_{i>0}(1-x^i)).
%C 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).
%H Alois P. Heinz, <a href="/A103650/b103650.txt">Table of n, a(n) for n = 1..500</a>
%F a(n) = Sum_{k>0} k * A264404(n,k). - _Alois P. Heinz_, Nov 29 2015
%F For n>2, a(n) is the Euler transform of [1,3,1,1,1,1,...]. - _Benedict W. J. Irwin_, Jul 29 2016
%F a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2). - _Vaclav Kotesovec_, Jul 30 2016
%e 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.
%t 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 *)
%t Table[Sum[PartitionsP[k]*(n-k)*(1 + (-1)^(n-k))/4, {k, 0, n}], {n, 1, 50}] (* _Vaclav Kotesovec_, Jul 30 2016 *)
%Y Cf. A014153.
%K easy,nonn
%O 1,4
%A _Vladeta Jovovic_, Mar 26 2005
%E More terms from _Robert G. Wilson v_, Mar 29 2005
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