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

5,5

Links

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

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

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 *)

Crossrefs

Column k=5 of A197126.

Sequence in context: A091608 A317081 A183562 * A241381 A237365 A322770

Adjacent sequences: A222702 A222703 A222704 * A222706 A222707 A222708

Keyword

nonn

Author

Alois P. Heinz, Feb 28 2013

Status

approved