A336516 Sum of parts, counted without multiplicity, in all compositions of n.

0, 1, 3, 10, 24, 59, 136, 309, 682, 1493, 3223, 6904, 14675, 31013, 65202, 136512, 284748, 592082, 1227709, 2539516, 5241640, 10798133, 22206568, 45597489, 93495667, 191464970, 391636718, 800233551, 1633530732, 3331568080, 6789078236, 13824212219, 28129459098

Offset: 0

Author

Alois P. Heinz, Jul 24 2020

Keywords

• nonn

Examples

			a(4) = 1 + 1 + 2 + 1 + 2 + 1 + 2 + 2 + 1 + 3 + 3 + 1 + 4 = 24: (1)111, (1)1(2), (1)(2)1, (2)(1)1, (2)2, (1)(3), (3)(1), (4).
		

Links

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

Crossrefs

Cf. A001787 (all parts), A014153 (the same for partitions), A336511, A336512, A336579, A336875.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
      `if`(i<1, 0, add((p-> [0, `if`(j=0, 0, p[1]*i)]+p)(
         b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..38);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
     If[i < 1, {0, 0}, Sum[Function[{0, If[j == 0, 0, #[[1]]*i]} + #][
       b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)