A064311 Generalized Catalan numbers C(-2; n).

1, 1, -1, 5, -25, 141, -849, 5349, -34825, 232445, -1582081, 10938709, -76616249, 542472685, -3876400305, 27919883205, -202480492905, 1477306676445, -10836099051105, 79861379898165, -591082795606425

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

a(n) ~ -(-1)^n * 2^(3*n+1) / (25 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 03 2019

Mathematica

a[n_] := If[n==0, 1, Sum[(n-m)*Binomial[n+m-1, m]*(-2)^m/n, {m, 0, n-1}]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 03 2019 *)

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: A094094 A344249 A081683 * A122441 A114870 A222676

Adjacent sequences: A064308 A064309 A064310 * A064312 A064313 A064314

Keyword

sign,easy

Author

Wolfdieter Lang, Sep 21 2001

Status

approved