A318114 Number of compositions of n into exactly n nonnegative parts <= six.

1, 1, 3, 10, 35, 126, 462, 1709, 6371, 23905, 90178, 341705, 1299662, 4958968, 18973111, 72763785, 279636451, 1076635399, 4151948115, 16035014604, 62009441410, 240083933750, 930547077155, 3610295962089, 14019766334990, 54487995870126, 211931334450696

Offset: 0

Author

Alois P. Heinz, Aug 17 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1675

Crossrefs

Column k=6 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, 6))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Formula

a(n) = [x^n] ((x^7-1)/(x-1))^n.

a(n) <= A088218(n) with equality only for n < 7.

From Peter Bala, Mar 31 2020: (Start)

a(n) = Sum_{i=0..n/7} (-1)^i*C(n,i)*C(2*n-7*i-1,n-7*i).

a(p) == 1 (mod p^2) for any prime p > 7.

More generally, we may have a(p^k) == a(p^(k-1)) (mod p^(2*k)) for k >= 2 and any prime p.

The sequence b(n) := [x^n] ( F(x)/F(-x) )^n, where F(x) = (x^7 - 1)/(x - 1), may satisfy the stronger congruences b(p) == 2 (mod p^3) for prime p > 7 (checked up to p = 499). (End)