cp's OEIS Frontend

A222705 Total number of parts of multiplicity 5 in all partitions of n.

%I A222705 #10 May 24 2018 04:22:58
%S A222705 1,0,1,1,2,3,5,5,9,11,17,21,31,37,53,67,90,113,151,186,246,305,392,
%T A222705 486,620,762,962,1181,1473,1802,2235,2716,3345,4056,4956,5990,7283,
%U A222705 8759,10598,12709,15297,18283,21917,26099,31165,37009,44014,52113,61776,72918
%N A222705 Total number of parts of multiplicity 5 in all partitions of n.
%H A222705 Alois P. Heinz, <a href="/A222705/b222705.txt">Table of n, a(n) for n = 5..1000</a>
%F A222705 G.f.: (x^5/(1-x^5)-x^6/(1-x^6))/Product_{j>0}(1-x^j).
%F A222705 a(n) ~ exp(Pi*sqrt(2*n/3)) / (60*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, May 24 2018
%p A222705 b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A222705       add((l->`if`(m=5, l+[0, l[1]], l))(b(n-p*m, p-1)), m=0..n/p)))
%p A222705     end:
%p A222705 a:= n-> b(n, n)[2]:
%p A222705 seq(a(n), n=5..60);
%t A222705 b[n_, p_] := b[n, p] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[Function[l, If[m == 5, l + {0, l[[1]]}, l]][b[n - p*m, p - 1]], {m, 0, n/p}]]];
%t A222705 a[n_] := b[n, n][[2]];
%t A222705 Table[a[n], {n, 5, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A222705 Column k=5 of A197126.
%K A222705 nonn
%O A222705 5,5
%A A222705 _Alois P. Heinz_, Feb 28 2013