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

0,4

Comments

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

Links

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

Formula

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

Example

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.

Maple

g:= proc(n, i, t) `if`(n=0, `if`(i=0, t!, 0),
     `if`(i<1 or n<i*(i+1)/2, 0, b(n, i, t)))
    end:
b:= proc(n, i, t) option remember;
      add(g(n-i*j, i-1, t+j)/j!, j=1..n/i)
    end:
a:= 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 *)

Crossrefs

Cf. A003056, A107429, A373118, A373306.

Sequence in context: A076860 A283592 A067589 * A059613 A242503 A116048

Adjacent sequences: A373302 A373303 A373304 * A373306 A373307 A373308

Keyword

nonn

Author

Alois P. Heinz, May 31 2024

Status

approved