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

0,3

Comments

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

Links

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

Formula

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

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

Example

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).

Maple

b:= proc(n, i, p) option remember; `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)))
    end:
a:= n-> `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 *)

Crossrefs

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

Sequence in context: A049818 A066189 A278834 * A306906 A174658 A326300

Adjacent sequences: A336893 A336894 A336895 * A336897 A336898 A336899

Keyword

nonn

Author

Alois P. Heinz, Aug 07 2020

Status

approved