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A064092 Generalized Catalan numbers C(9; n). 5
1, 1, 10, 181, 4078, 102826, 2777212, 78571837, 2298558934, 68964092542, 2110472708140, 65620725560578, 2067160250751436, 65833929303952564, 2116166898185821792, 68565914052628406221, 2237022199842087256678 (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=9, beta =1 (or alpha=1, beta=9).

LINKS

Table of n, a(n) for n=0..16.

FORMULA

G.f.: (1+9*x*c(9*x)/8)/(1+x/8) = 1/(1-x*c(9*x)) with c(x) g.f. of Catalan numbers A000108.

a(n)= sum((n-m)*binomial(n-1+m, m)*(9^m)/n, m=0..n-1) = ((-1/8)^n)*(1-9*sum(C(k)*(-72)^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)*9^(n-k) } . - Philippe Deléham, Jan 19 2004

Conjecture: 8*n*a(n) +(-287*n+432)*a(n-1) +18*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

MATHEMATICA

a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*9^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 09 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-8*x^2)/(1+x)^2+O(x^(n+1))), n)) /* _Ralf Stephan+ */

CROSSREFS

A064091 (C(8, n)).

Sequence in context: A318796 A054918 A095807 * A171513 A240405 A304936

Adjacent sequences:  A064089 A064090 A064091 * A064093 A064094 A064095

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 13 2001

STATUS

approved

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Last modified October 20 04:37 EDT 2018. Contains 316378 sequences. (Running on oeis4.)