A349302 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^6)).
1, 0, 1, 6, 43, 321, 2500, 20096, 165621, 1392397, 11896823, 103014141, 902035660, 7974080834, 71070247438, 637937825112, 5761970031357, 52329993278856, 477588786637264, 4377832437503643, 40288077072190109, 372086539388626537, 3447632819399550915
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..500
Programs
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Mathematica
nmax = 22; 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[(-1)^(n - k) Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 22}]
Formula
a(n) = Sum_{k=0..n} (-1)^(n-k) * 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 = 0.1008057775745727124639860500770912830001828593281202101426766... is the root of the equation 7^7 * r = 6^6 * (1+r)^6. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^6), where F(x) = Sum_{n >= 0} A002296(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^7. (End)