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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. 6
1, 5, 45, 505, 6345, 85405, 1204245, 17558705, 262577745, 4005148405, 62070886845, 974612606505, 15471084667545, 247876665109005, 4003225107031845, 65101209768055905, 1065128963164067745, 17520376884067071205, 289572455530026439245, 4806489064223483202905 (list; graph; refs; listen; history; text; internal format)
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_{0<=k<=n} A088617(n,k)*4^k.

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

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

a(0) = 1, a(n) = a(n-1)+4*Sum_{0<=k<=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

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

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)