A320373 Number of parts in all partitions of n with largest multiplicity three.

3, 0, 4, 7, 13, 14, 27, 33, 55, 72, 107, 137, 196, 250, 344, 442, 588, 750, 982, 1234, 1591, 1992, 2523, 3135, 3944, 4857, 6035, 7408, 9121, 11109, 13599, 16465, 20004, 24122, 29112, 34927, 41952, 50078, 59836, 71169, 84625, 100219, 118716, 140061, 165225

Offset: 3

Author

Alois P. Heinz, Oct 11 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..5000

Crossrefs

Column k=3 of A213177.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((l-> [0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(3):
seq(a(n), n=3..60);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0},        Sum[Function[l, {0, l[[1]] j} + l][b[n - i j, i - 1, k]], {j, 0, Min[n/i, k]}]]];
a[n_] := With[{k = 3}, (b[n, n, k] - b[n, n, k - 1])[[2]]];
a /@ Range[3, 60] (* Jean-François Alcover, Dec 13 2020, after Alois P. Heinz *)

Formula

a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 25 2018