A349292 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^6)).

1, 2, 15, 190, 2871, 47643, 838888, 15389452, 290951545, 5629024955, 110908062511, 2217739684483, 44891645810124, 918086053852234, 18941156419798530, 393742848618632760, 8239112912485293357, 173406208518520952066, 3668419587671991125142

Offset: 0

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A002296, A007317, A199475, A346649, A349289, A349290, A349291, A349293.

Programs

Mathematica

nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^6)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(7*k,k) / (6*k+1).

a(n) ~ sqrt(1 + 5*r) / (2 * 7^(2/3) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/6)), where r = 0.043408935906208378827553096713877784793679356... is the root of the equation 7^7 * r = 6^6 * (1-r)^6. - Vaclav Kotesovec, Nov 14 2021

a(n) = 1 + Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 11 2025