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

3,4

Links

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

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

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 *)

Crossrefs

Column k=3 of A222730.

Sequence in context: A368163 A202816 A089411 * A025508 A023189 A067137

Adjacent sequences: A222728 A222729 A222730 * A222732 A222733 A222734

Keyword

nonn

Author

Alois P. Heinz, Mar 03 2013

Status

approved