A336896 Sum of the leftmost parts in all compositions of n into distinct parts.

0, 1, 2, 6, 8, 15, 30, 42, 64, 99, 190, 242, 384, 533, 798, 1380, 1824, 2635, 3762, 5320, 7280, 12327, 15554, 22632, 30720, 43425, 57538, 80730, 122920, 159239, 220830, 299150, 406656, 542883, 733278, 962710, 1443600, 1820437, 2496638, 3280992, 4451120

Offset: 0

Author

Alois P. Heinz, Aug 07 2020

Keywords

• nonn

Comments

Also sum of the rightmost parts in all compositions of n into distinct parts.

Examples

			a(6) = 30 = 1 + 1 + 2 + 2 + 3 + 3 + 2 + 4 + 1 + 5 + 6: (1)23, (1)32, (2)13, (2)31, (3)12, (3)21, (2)4, (4)2, (1)5, (5)1, (6).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000225 (the same for all compositions), A008289, A032020, A032153.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 `if`(n=0, 0, n*b(n$2, -1)):
seq(a(n), n=0..50);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[i*(i + 1)/2 < n, 0,
     If[n == 0, p!, b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p + 1]]];
a[n_] := If[n == 0, 0, n*b[n, n, -1]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 13 2022, after Alois P. Heinz *)

Formula

a(n) = n * Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} (k-1)! * A008289(n,k).

a(n) = n * A032153(n).