A352119 Expansion of e.g.f. 1/(2 - exp(4*x))^(1/4).

1, 1, 9, 121, 2289, 56401, 1713849, 61939081, 2595199329, 123690992161, 6608289658089, 391154820258841, 25408740616159569, 1797051730819428721, 137463201511019813529, 11308020549364112399401, 995455518982520306979009, 93373681491447943767190081

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).

a(n) ~ n! * 2^(2*n - 1/4) / (Gamma(1/4) * n^(3/4) * log(2)^(n + 1/4)). - Vaclav Kotesovec, Mar 05 2022

From Seiichi Manyama, Nov 18 2023: (Start)

a(0) = 1; a(n) = Sum_{k=1..n} 4^k * (1 - 3/4 * k/n) * binomial(n,k) * a(n-k).

a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-4)^k * binomial(n-1,k) * a(n-k). (End)

Mathematica

m = 17; Range[0, m]! * CoefficientList[Series[(2 - Exp[4*x])^(-1/4), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(4*x))^(1/4)))
(PARI) a(n) = sum(k=0, n, 4^(n-k)*prod(j=0, k-1, 4*j+1)*stirling(n, k, 2));

Crossrefs

Cf. A000670, A352117, A352118.

Cf. A352073.

Sequence in context: A321847 A050353 A112941 * A258380 A360547 A045976

Adjacent sequences: A352116 A352117 A352118 * A352120 A352121 A352122

Keyword

nonn

Author

Seiichi Manyama, Mar 05 2022

Status

approved