A373305 Sum over all complete compositions of n of the element set cardinality.

0, 1, 1, 5, 7, 15, 41, 77, 161, 325, 727, 1460, 3058, 6228, 12815, 26447, 54099, 110800, 226247, 461531, 939678, 1914189, 3890279, 7905962, 16045367, 32550830, 65971827, 133645098, 270561031, 547468214, 1107208235, 2238242852, 4522679064, 9135128917

Offset: 0

Author

Alois P. Heinz, May 31 2024

Keywords

• nonn

Comments

A complete composition of n has element set [k] with k<=n (without gaps).

Examples

			a(1) = 1: 1.
a(2) = 1: 11.
a(3) = 5 = 2 + 2 + 1: 12, 21, 111.
a(4) = 7 = 2 + 2 + 2 + 1: 112, 121, 211, 1111.
a(5) = 15 = 7*2 + 1: 122, 212, 221, 1112, 1121, 1211, 2111, 11111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A003056, A107429, A373118, A373306.

Programs

Maple

g:= proc(n, i, t) `if`(n=0, `if`(i=0, t!, 0),
     `if`(i<1 or n add(g(n, k, 0)*k, k=0..floor((sqrt(1+8*n)-1)/2)):
seq(a(n), n=0..33);

Mathematica

g[n_, i_, t_] := If[n == 0, If[i == 0, t!, 0], If[i < 1 || n < i*(i+1)/2, 0, b[n, i, t]]];
b[n_, i_, t_] := b[n, i, t] = Sum[g[n-i*j, i-1, t+j]/j!, {j, 1, n/i}];
a[n_] := Sum[g[n, k, 0]*k, {k, 0, Floor[(Sqrt[1 + 8*n] - 1)/2]}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jun 08 2024, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..A003056(n)} k * A373118(n,k).