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A222704 Total number of parts of multiplicity 4 in all partitions of n.

%I A222704 #12 May 24 2018 04:22:36
%S A222704 1,0,1,1,3,3,5,6,11,13,20,24,37,45,64,80,110,137,184,229,303,375,486,
%T A222704 602,772,951,1202,1478,1853,2267,2817,3432,4236,5142,6300,7620,9284,
%U A222704 11185,13553,16273,19625,23478,28187,33613,40192,47778,56904,67443,80051
%N A222704 Total number of parts of multiplicity 4 in all partitions of n.
%H A222704 Alois P. Heinz, <a href="/A222704/b222704.txt">Table of n, a(n) for n = 4..1000</a>
%F A222704 G.f.: (x^4/(1-x^4)-x^5/(1-x^5))/Product_{j>0}(1-x^j).
%F A222704 a(n) ~ exp(Pi*sqrt(2*n/3)) / (40*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, May 24 2018
%p A222704 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A222704       add((l->`if`(m=4, l+[0, l[1]], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A222704     end:
%p A222704 a:= n-> b(n, n)[2]:
%p A222704 seq(a(n), n=4..60);
%t A222704 b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p<1, {0, 0}, Sum[Function[l, If[m == 4, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
%t A222704 a[n_] := b[n, n][[2]];
%t A222704 Table[a[n], {n, 4, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A222704 Column k=4 of A197126.
%K A222704 nonn
%O A222704 4,5
%A A222704 _Alois P. Heinz_, Feb 28 2013