A064327 Generalized Catalan numbers C(-5; n).

1, 1, -4, 41, -514, 7206, -108174, 1700721, -27646234, 460887086, -7836596944, 135380098426, -2369445113804, 41925242220616, -748729419265314, 13478117036893281, -244306305241572474, 4455242518055441046, -81683397232911983784, 1504758636166747742286

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..765

Crossrefs

Cf. A064334.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (11 +Sqrt(1+20*x))/(2*(6-x)) )); // G. C. Greubel, May 03 2019

Mathematica

CoefficientList[Series[(11 +Sqrt[1+20*x])/(2*(6-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PARI

my(x='x+O('x^30)); Vec((11 +sqrt(1+20*x))/(2*(6-x))) \\ G. C. Greubel, May 03 2019

Sage

def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], -5).simplify()
[a(n) for n in range(24)] # Peter Luschny, Nov 30 2014

Sage

((11 +sqrt(1+20*x))/(2*(6-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

Formula

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-5)^m/n.

a(n) = (1/6)^n*(1 + 5*Sum_{k=0..n-1} C(k)*(-5*6)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

G.f.: (1+5*x*c(-5*x)/6)/(1-x/6) = 1/(1-x*c(-5*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = hypergeometric([1-n, n], [-n], -5) for n > 0. - Peter Luschny, Nov 30 2014