A133305 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*4^i*5^(n-i), a(0) = 1.

1, 5, 45, 505, 6345, 85405, 1204245, 17558705, 262577745, 4005148405, 62070886845, 974612606505, 15471084667545, 247876665109005, 4003225107031845, 65101209768055905, 1065128963164067745, 17520376884067071205, 289572455530026439245, 4806489064223483202905

Offset

0,2

Comments

Fifth column of array A103209.

The Hankel transform of this sequence is 20^C(n+1,2). - Philippe Deléham, Oct 28 2007

Links

G. C. Greubel, Table of n, a(n) for n = 0..795

Samuele Giraudo, Operads from posets and Koszul duality, arXiv:1504.04529 [math.CO], 2015-2016.

Formula

G.f.: (1-z-sqrt(z^2-18*z+1))/(8*z).

a(n) = Sum_{k=0..n} A088617(n,k)*4^k.

a(n) = Sum_{k=0..n} A060693(n,k)*4^(n-k).

a(n) = Sum_{k=0..n} C(n+k, 2k)*4^k*C(k), C(n) given by A000108.

a(0) = 1, a(n) = a(n-1) + 4*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007

Conjecture: (n+1)*a(n) + 9*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014

G.f.: 1/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

a(n) = hypergeom([-n, n + 1], [2], -4]). - Peter Luschny, Jan 08 2018

a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 21 2021

Mathematica

a[n_] := Hypergeometric2F1[-n, n + 1, 2, -4];
Table[a[n], {n, 0, 16}] (* Peter Luschny, Jan 08 2018 *)
CoefficientList[Series[(1-x-Sqrt[x^2-18*x+1])/(8*x), {x, 0, 50}], x] (* G. C. Greubel, Feb 10 2018 *)

Prog

(PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-18*x+1))/(8*x)) \\ G. C. Greubel, Feb 10 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-18*x+1))/(8*x))) // G. C. Greubel, Feb 10 2018

Crossrefs

Cf. A000108, A060693, A103209, A103210, A103211.

Sequence in context: A151831 A233834 A188267 * A316705 A248586 A275576

Adjacent sequences: A133302 A133303 A133304 * A133306 A133307 A133308

Keyword

nonn

Author

Philippe Deléham, Oct 18 2007

Status

approved