A222705 - OEIS

%I #10 May 24 2018 04:22:58

%S 1,0,1,1,2,3,5,5,9,11,17,21,31,37,53,67,90,113,151,186,246,305,392,

%T 486,620,762,962,1181,1473,1802,2235,2716,3345,4056,4956,5990,7283,

%U 8759,10598,12709,15297,18283,21917,26099,31165,37009,44014,52113,61776,72918

%N Total number of parts of multiplicity 5 in all partitions of n.

%H Alois P. Heinz, <a href="/A222705/b222705.txt">Table of n, a(n) for n = 5..1000</a>

%F G.f.: (x^5/(1-x^5)-x^6/(1-x^6))/Product_{j>0}(1-x^j).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (60*Pi*sqrt(2*n)). - _Vaclav Kotesovec_, May 24 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]], 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..60);

%t 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 a[n_] := b[n, n][[2]];

%t Table[a[n], {n, 5, 60}] (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)

%Y Column k=5 of A197126.

%K nonn

%O 5,5

%A _Alois P. Heinz_, Feb 28 2013