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A222707 Total number of parts of multiplicity 7 in all partitions of n.

%I A222707 #10 May 24 2018 04:23:37
%S A222707 1,0,1,1,2,2,4,5,8,9,14,17,25,30,43,53,72,88,118,145,190,234,301,370,
%T A222707 471,575,724,884,1102,1339,1657,2007,2465,2975,3630,4369,5301,6355,
%U A222707 7672,9171,11018,13126,15706,18655,22231,26329,31260,36920,43688,51454,60693
%N A222707 Total number of parts of multiplicity 7 in all partitions of n.
%H A222707 Alois P. Heinz, <a href="/A222707/b222707.txt">Table of n, a(n) for n = 7..1000</a>
%F A222707 G.f.: (x^7/(1-x^7)-x^8/(1-x^8))/Product_{j>0}(1-x^j).
%F A222707 a(n) ~ exp(Pi*sqrt(2*n/3)) / (112*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, May 24 2018
%p A222707 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A222707       add((l->`if`(m=7, l+[0, l[1]], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A222707     end:
%p A222707 a:= n-> b(n, n)[2]:
%p A222707 seq(a(n), n=7..60);
%t A222707 b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 7, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
%t A222707 a[n_] := b[n, n][[2]];
%t A222707 Table[a[n], {n, 7, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A222707 Column k=7 of A197126.
%K A222707 nonn
%O A222707 7,5
%A A222707 _Alois P. Heinz_, Feb 28 2013