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A222732 Total sum of parts of multiplicity 4 in all partitions of n.

%I A222732 #10 May 29 2018 09:19:50
%S A222732 1,0,1,1,4,4,6,8,16,19,30,36,59,73,106,135,191,242,331,420,569,712,
%T A222732 941,1183,1546,1931,2476,3087,3933,4872,6137,7568,9471,11629,14427,
%U A222732 17647,21758,26499,32470,39393,48030,58028,70385,84749,102348,122794,147633,176554
%N A222732 Total sum of parts of multiplicity 4 in all partitions of n.
%H A222732 Alois P. Heinz, <a href="/A222732/b222732.txt">Table of n, a(n) for n = 4..1000</a>
%F A222732 G.f.: (x^4/(1-x^4)^2-x^5/(1-x^5)^2)/Product_{i>=1}(1-x^i).
%F A222732 a(n) ~ 9 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (800 * Pi^2). - _Vaclav Kotesovec_, May 29 2018
%p A222732 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A222732       add((l->`if`(m=4, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A222732     end:
%p A222732 a:= n-> b(n, n)[2]:
%p A222732 seq(a(n), n=4..55);
%t A222732 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][[6]]; Table[a[n], {n, 4, 55}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)
%Y A222732 Column k=4 of A222730.
%K A222732 nonn
%O A222732 4,5
%A A222732 _Alois P. Heinz_, Mar 03 2013