A349301 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^5)).
1, 0, 1, 5, 31, 200, 1351, 9430, 67531, 493505, 3665981, 27602081, 210179437, 1615820402, 12524590873, 97775503808, 768083233899, 6067097140799, 48159634951855, 383965003803985, 3073379977522321, 24688458872260007, 198968304164411309
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]^5)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Table[Sum[(-1)^(n - k) Binomial[n + 4 k, 5 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 22}]
Formula
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+4*k,5*k) * binomial(6*k,k) / (5*k+1).
a(n) ~ sqrt(1 - 4*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi*(1+r)) * n^(3/2) * r^(n + 1/5)), where r = 0.11589193448796656683553561932282471511616945026903125324... is the real root of the equation 6^6 * r = 5^5 * (1+r)^5. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^5), where F(x) = Sum_{n >= 0} A002295(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^6. (End)