A222731 Total sum of parts of multiplicity 3 in all partitions of n.

1, 0, 1, 3, 4, 4, 11, 13, 21, 30, 44, 59, 92, 115, 165, 225, 305, 394, 546, 700, 931, 1204, 1572, 2005, 2613, 3290, 4218, 5328, 6745, 8423, 10630, 13193, 16475, 20386, 25269, 31072, 38346, 46882, 57478, 70066, 85415, 103582, 125794, 151916, 183576, 220962

Offset: 3

Author

Alois P. Heinz, Mar 03 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A222730.

Programs

Maple

b:= proc(n, p) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
      add((l->`if`(m=3, l+[0, l[1]*p], l))(b(n-p*m, p-1)), m=0..n/p)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=3..50);

Mathematica

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[5]]; Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

Formula

G.f.: (x^3/(1-x^3)^2-x^4/(1-x^4)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 7 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (288 * Pi^2). - Vaclav Kotesovec, May 29 2018