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A064090 Generalized Catalan numbers C(7; n). 4
1, 1, 8, 113, 1982, 38886, 817062, 17981769, 409186310, 9549411950, 227307541448, 5497312072330, 134696099554276, 3336563455537768, 83419226227330722, 2102274863070771033, 53347639317495439302 (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=7, beta =1 (or alpha=1, beta=7).

LINKS

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

FORMULA

G.f.: (1+7*x*c(7*x)/6)/(1+x/6) = 1/(1-x*c(7*x)) with c(x) g.f. of Catalan numbers A000108.

a(n)= sum((n-m)*binomial(n-1+m, m)*(7^m)/n, m=0..n-1) = ((-1/6)^n)*(1-7*sum(C(k)*(-42)^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)*7^(n-k) } . - Philippe Deléham, Jan 19 2004

Conjecture: 6*n*a(n) +(-167*n+252)*a(n-1) +14*(-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]*7^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-6*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

CROSSREFS

A064089 (C(6, n)).

Sequence in context: A164774 A259590 A155460 * A072402 A092084 A099715

Adjacent sequences:  A064087 A064088 A064089 * A064091 A064092 A064093

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 13 2001

STATUS

approved

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Last modified October 17 07:05 EDT 2018. Contains 316276 sequences. (Running on oeis4.)