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A064311 Generalized Catalan numbers C(-2; n). 4
1, 1, -1, 5, -25, 141, -849, 5349, -34825, 232445, -1582081, 10938709, -76616249, 542472685, -3876400305, 27919883205, -202480492905, 1477306676445, -10836099051105, 79861379898165, -591082795606425 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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)*((-2)^m)/n, m=0..n-1) = ((1/3)^n)*(1+2*sum(C(k)*(-2*3)^k, k=0..n-1)), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+2*x*c(-2*x)/3)/(1-x/3) = 1/(1-x*c(-2*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = hypergeom([1-n, n], [-n], -2) for n>0. - Peter Luschny, Nov 30 2014

PROG

(Sage)

import mpmath

mp.dps = 25; mp.pretty = True

a = lambda n: mpmath.hyp2f1(1-n, n, -n, -2) if n>0 else 1

[int(a(n)) for n in range(21)] # Peter Luschny, Nov 30 2014

CROSSREFS

Cf. A064062.

Sequence in context: A124891 A094094 A081683 * A122441 A114870 A222676

Adjacent sequences:  A064308 A064309 A064310 * A064312 A064313 A064314

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Sep 21 2001

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)