A222732 Total sum of parts of multiplicity 4 in all partitions of n.

1, 0, 1, 1, 4, 4, 6, 8, 16, 19, 30, 36, 59, 73, 106, 135, 191, 242, 331, 420, 569, 712, 941, 1183, 1546, 1931, 2476, 3087, 3933, 4872, 6137, 7568, 9471, 11629, 14427, 17647, 21758, 26499, 32470, 39393, 48030, 58028, 70385, 84749, 102348, 122794, 147633, 176554

Offset

4,5

Links

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Formula

G.f.: (x^4/(1-x^4)^2-x^5/(1-x^5)^2)/Product_{i>=1}(1-x^i).

a(n) ~ 9 * sqrt(3) * exp(Pi*sqrt(2*n/3)) / (800 * 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=4, 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=4..55);

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][[6]]; Table[a[n], {n, 4, 55}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

Crossrefs

Column k=4 of A222730.

Sequence in context: A175216 A053639 A100090 * A363703 A074161 A201401

Adjacent sequences: A222729 A222730 A222731 * A222733 A222734 A222735

Keyword

nonn

Author

Alois P. Heinz, Mar 03 2013

Status

approved