A367237 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x)^2)^2.

1, 1, 4, 20, 114, 702, 4550, 30585, 211270, 1490561, 10695354, 77809481, 572608270, 4254996670, 31882486314, 240620654468, 1827464108766, 13956516915303, 107114560278680, 825727777034002, 6390721805005678, 49638977802126104, 386824024893533450

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).

a(n) ~ sqrt((12735 + (849*(23867603343 - 274945024*sqrt(849)))^(1/3) + (849*(23867603343 + 274945024*sqrt(849)))^(1/3))/283) * ((2053 + (10379182717 - 43903488*sqrt(849))^(1/3) + (10379182717 + 43903488*sqrt(849))^(1/3))^n / (sqrt(Pi) * n^(3/2) * 2^(8*n + 9/2) * 3^(n + 1/2))). - Vaclav Kotesovec, Nov 13 2023

Mathematica

CoefficientList[Series[Root[-1 + #1 + x*#1^2 - 2*x*#1^3 - x^2*#1^4 + x^2*#1^5&, 1], {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 13 2023 *)

Prog

(PARI) a(n, s=2, t=2, u=2) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));

Crossrefs

Cf. A001764, A367236, A367238.

Sequence in context: A362223 A108447 A287512 * A211248 A346414 A028475

Adjacent sequences: A367234 A367235 A367236 * A367238 A367239 A367240

Keyword

nonn

Author

Seiichi Manyama, Nov 11 2023

Status

approved