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

1, 0, 1, 1, 2, 3, 5, 5, 9, 11, 17, 21, 31, 37, 53, 67, 90, 113, 151, 186, 246, 305, 392, 486, 620, 762, 962, 1181, 1473, 1802, 2235, 2716, 3345, 4056, 4956, 5990, 7283, 8759, 10598, 12709, 15297, 18283, 21917, 26099, 31165, 37009, 44014, 52113, 61776, 72918

Offset: 5

Author

Alois P. Heinz, Feb 28 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..1000

Crossrefs

Column k=5 of A197126.

Programs

Maple

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=5, l+[0, l[1]], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=5..60);

Mathematica

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}]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 5, 60}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Formula

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

a(n) ~ exp(Pi*sqrt(2*n/3)) / (60*Pi*sqrt(2*n)). - Vaclav Kotesovec, May 24 2018