OFFSET
0,4
COMMENTS
See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
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
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
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Sep 21 2001
STATUS
approved