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A064091 Generalized Catalan numbers C(8; n). 4
1, 1, 9, 145, 2905, 65121, 1563561, 39322929, 1022586105, 27272680705, 741894295369, 20504949587409, 574176887116441, 16254518495907745, 464436319229036265, 13376293681432402545, 387925710986712480825 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=8, beta =1 (or alpha=1, beta=8).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

G.f.: (1+8*x*c(8*x)/7)/(1+x/7) = 1/(1-x*c(8*x)) with c(x) g.f. of Catalan numbers A000108.

a(n)=sum((n-m)*binomial(n-1+m, m)*(8^m)/n, m=0..n-1) = ((-1/7)^n)*(1-8*sum(C(k)*(-56)^k, k=0..n-1)), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

a(n) = Sum{ k= 0...n, A059365(n, k)*8^(n-k) } . - Philippe Deléham, Jan 19 2004

Conjecture: 7*n*a(n) +(-223*n+336)*a(n-1) +16*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

a(n) ~ 2^(5*n+3)/(225*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

MATHEMATICA

a[0] = 1; a[n_] := Sum[(n-m)*Binomial[n+m-1, m]*(8^m)/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 21 2013 *)

Table[FullSimplify[(-1)^(2*n)*2^(3+5*n)*(1/2*(2*n-1))! Hypergeometric2F1[1, 1/2+n, 2+n, -224]/(Sqrt[Pi]*(n+1)!)], {n, 0, 20}] (* Vaclav Kotesovec, Aug 13 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-7*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

CROSSREFS

A064090 (C(7, n)).

Sequence in context: A173213 A223371 A046529 * A132060 A320333 A178185

Adjacent sequences:  A064088 A064089 A064090 * A064092 A064093 A064094

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 13 2001

STATUS

approved

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Last modified October 18 05:17 EDT 2018. Contains 316304 sequences. (Running on oeis4.)