A318115 Number of compositions of n into exactly n nonnegative parts <= seven.

1, 1, 3, 10, 35, 126, 462, 1716, 6427, 24229, 91828, 349570, 1335698, 5119856, 19678452, 75814560, 292695291, 1132074847, 4385740683, 17015510820, 66102536360, 257103599280, 1001078753370, 3901733646660, 15220908863866, 59427261280126, 232201593611796

Offset: 0

Author

Alois P. Heinz, Aug 17 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1669

Crossrefs

Column k=7 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, 7))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Formula

a(n) = [x^n] ((x^8-1)/(x-1))^n.

a(n) <= A088218(n) with equality only for n < 8.

a(n) = Sum_{k=0..floor(n/8)} (-1)^k * binomial(n,k) * binomial(2*n-8*k-1,n-8*k). - Ilya Gutkovskiy, Nov 03 2021