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

0,4

Links

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

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)

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}]

Crossrefs

Cf. A002295, A005043, A346627, A346666, A349291, A349299, A349300, A349302, A349303.

Sequence in context: A260782 A084235 A356290 * A288688 A002469 A296032

Adjacent sequences: A349298 A349299 A349300 * A349302 A349303 A349304

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Nov 13 2021

Status

approved