cp's OEIS Frontend

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

%I A318114 #16 Oct 06 2021 14:29:58
%S A318114 1,1,3,10,35,126,462,1709,6371,23905,90178,341705,1299662,4958968,
%T A318114 18973111,72763785,279636451,1076635399,4151948115,16035014604,
%U A318114 62009441410,240083933750,930547077155,3610295962089,14019766334990,54487995870126,211931334450696
%N A318114 Number of compositions of n into exactly n nonnegative parts <= six.
%H A318114 Alois P. Heinz, <a href="/A318114/b318114.txt">Table of n, a(n) for n = 0..1675</a>
%F A318114 a(n) = [x^n] ((x^7-1)/(x-1))^n.
%F A318114 a(n) <= A088218(n) with equality only for n < 7.
%F A318114 From _Peter Bala_, Mar 31 2020: (Start)
%F A318114 a(n) = Sum_{i=0..n/7} (-1)^i*C(n,i)*C(2*n-7*i-1,n-7*i).
%F A318114 a(p) == 1 (mod p^2) for any prime p > 7.
%F A318114 More generally, we may have a(p^k) == a(p^(k-1)) (mod p^(2*k)) for k >= 2 and any prime p.
%F A318114 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)
%p A318114 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A318114       `if`(i=0, 0, add(b(n-j, i-1), j=0..min(n, 6))))
%p A318114     end:
%p A318114 a:= n-> b(n$2):
%p A318114 seq(a(n), n=0..30);
%Y A318114 Column k=6 of A305161.
%Y A318114 Cf. A088218.
%K A318114 nonn
%O A318114 0,3
%A A318114 _Alois P. Heinz_, Aug 17 2018