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

Author

Wolfdieter Lang, Sep 21 2001

Keywords

• sign
• easy

Comments

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

Crossrefs

Generalized Catalan numbers C(m; n): A000012 (m = 0), A000108 (m = 1), A064062 (m = 2), A064063 (m = 3), A064087 - A064093 (m = 4 thru 10); A064310 (m = -1) and A064325 - A064333 (m = -3 thru -11).

Programs

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 *)

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

Formula

a(n) = (1/n) * Sum_{m = 0..n-1} (n-m)*binomial(n-1+m, m)*(-2)^m = ((1/3)^n)*(1 + 2*Sum_{k = 0..n-1} C(k)*(-2*3)^k), for n >= 1, with a(0) := 1, and where C(n) = A000108(n), the Catalan numbers.

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

G.f. A(x) = 1 + series_reversion(x*(1 - (m-1)*x)/(1 + x)^2) at m = -2. - Peter Bala, Sep 08 2024