A349300 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^4)).

1, 0, 1, 4, 21, 114, 651, 3844, 23301, 144169, 906866, 5782350, 37289431, 242793439, 1593918916, 10538988984, 70121101825, 469133993094, 3154115695476, 21299373321344, 144402246424591, 982506791975780, 6706724412165956, 45917245477282994

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+3*k,4*k) * binomial(5*k,k) / (4*k+1).

a(n) = (-1)^5*F([1/5, 2/5, 3/5, 4/5, (1+n)/3, (2+n)/3, (3+n)/3, -n], [1/4, 1/2, 1/2, 3/4, 3/4, 1, 5/4], 3^3*5^5/2^16), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021

a(n) ~ sqrt(1 - 3*r) / (2 * 5^(3/4) * sqrt(2*Pi*(1+r)) * n^(3/2) * r^(n + 1/4)), where r = 0.136824361675510443450981569282313811786270109272790613523286... is the root of the equation 5^5 * r = 4^4 * (1+r)^4. - Vaclav Kotesovec, Nov 14 2021

From Peter Bala, Jun 02 2024: (Start)

A(x) = 1/(1 + x)*F(x/(1 + x)^4), where F(x) = Sum_{n >= 0} A002294(n)*x^n.

A(x) = 1/(1 + x) + x*A(x)^5. (End)

Mathematica

nmax = 23; A[_] = 0; Do[A[x_] = 1/((1 + x) (1 - x A[x]^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n + 3 k, 4 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 23}]

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n+3*k, 4*k) * binomial(5*k, k) / (4*k+1)); \\ Michel Marcus, Nov 14 2021

Crossrefs

Cf. A002294, A005043, A346627, A346665, A349290, A349299, A349301, A349302, A349303.

Sequence in context: A127111 A270787 A190089 * A240436 A255139 A015554

Adjacent sequences: A349297 A349298 A349299 * A349301 A349302 A349303

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Nov 13 2021

Status

approved