A318116 Number of compositions of n into exactly n nonnegative parts <= eight.

1, 1, 3, 10, 35, 126, 462, 1716, 6435, 24301, 92278, 351990, 1347710, 5176640, 19938348, 76977360, 297811491, 1154300103, 4481325903, 17423296059, 67830758310, 264387659490, 1031636761290, 4029420952890, 15752622069630, 61634789550126, 241342568718696

Offset: 0

Author

Alois P. Heinz, Aug 17 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1666

Crossrefs

Column k=8 of A305161.

Cf. A088218.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i=0, 0, add(b(n-j, i-1), j=0..min(n, 8))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Formula

a(n) = [x^n] ((x^9-1)/(x-1))^n.

a(n) <= A088218(n) with equality only for n < 9.

a(n) = Sum_{k=0..floor(n/9)} (-1)^k * binomial(n,k) * binomial(2*n-9*k-1,n-9*k). - Ilya Gutkovskiy, Nov 03 2021