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A064326 Generalized Catalan numbers C(-4; n). 2
1, 1, -3, 25, -251, 2817, -33843, 425769, -5537835, 73865617, -1004862179, 13888533561, -194475377243, 2752994728225, -39333541106835, 566464908534345, -8214515461250955, 119845125957958065, -1757855400878129475, 25906894146115000665, -383443906519878272955 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

LINKS

Table of n, a(n) for n=0..20.

FORMULA

a(n) = sum((n-m)*binomial(n-1+m, m)*((-4)^m)/n, m=0..n-1) = ((1/5)^n)*(1+4*sum(C(k)*(-4*5)^k, k=0..n-1)), n >= 1, a(0) = 1; with C(n) = A000108(n) (Catalan).

G.f.: (1+4*x*c(-4*x)/5)/(1-x/5) = 1/(1-x*c(-4*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = hypergeometric([1-n, n], [-n], -4) for n>0. - Peter Luschny, Nov 30 2014

PROG

(Sage)

def a(n):

    if n==0: return 1

    return hypergeometric([1-n, n], [-n], -4).simplify()

[a(n) for n in range(20)] # Peter Luschny, Nov 30 2014

CROSSREFS

Cf. A064334.

Sequence in context: A245925 A260209 A086655 * A006091 A009042 A080896

Adjacent sequences:  A064323 A064324 A064325 * A064327 A064328 A064329

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Sep 21 2001

STATUS

approved

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Last modified October 18 07:00 EDT 2018. Contains 316307 sequences. (Running on oeis4.)