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%I #10 May 29 2018 09:19:40
%S 1,0,1,1,2,4,6,6,11,14,23,29,43,52,76,100,135,174,235,294,397,500,651,
%T 821,1060,1324,1692,2107,2658,3297,4139,5089,6339,7778,9604,11746,
%U 14425,17533,21427,25960,31548,38080,46070,55375,66718,79957,95906,114555
%N Total sum of parts of multiplicity 5 in all partitions of n.
%H Alois P. Heinz, <a href="/A222733/b222733.txt">Table of n, a(n) for n = 5..1000</a>
%F G.f.: (x^5/(1-x^5)^2-x^6/(1-x^6)^2)/Product_{i>=1}(1-x^i).
%F a(n) ~ 11 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (1800 * Pi^2). - _Vaclav Kotesovec_, May 29 2018
%p b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p add((l->`if`(m=5, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
%p end:
%p a:= n-> b(n, n)[2]:
%p seq(a(n), n=5..55);
%t b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[7]]; Table[a[n], {n, 5, 55}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)
%Y Column k=5 of A222730.
%K nonn
%O 5,5
%A _Alois P. Heinz_, Mar 03 2013
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