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 A318114 Number of compositions of n into exactly n nonnegative parts <= six. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1675 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) 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); CROSSREFS Column k=6 of A305161. Cf. A088218. Sequence in context: A318113 A216710 A087946 * A122068 A099908 A363781 Adjacent sequences: A318111 A318112 A318113 * A318115 A318116 A318117 KEYWORD nonn AUTHOR Alois P. Heinz, Aug 17 2018 STATUS approved

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Last modified October 3 11:12 EDT 2023. Contains 365861 sequences. (Running on oeis4.)