A064325 Generalized Catalan numbers C(-3; n).

1, 1, -2, 13, -98, 826, -7448, 70309, -686090, 6865150, -70057772, 726325810, -7628741204, 81002393668, -868066319108, 9376806129493, -101988620430938, 1116026661667318, -12277755319108748, 135715825209716038, -1506587474535945788, 16789107646422189868, -187747069029477151328

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..850

Crossrefs

Cf. A064334, A000108.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (7 +Sqrt(1+12*x))/(2*(4-x)) )); // G. C. Greubel, May 03 2019

Mathematica

a[0] = 1;
a[n_] := Sum[(n-m) Binomial[n+m-1, m] (-3)^m/n, {m, 0, n-1}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jul 30 2018 *)
CoefficientList[Series[(7 +Sqrt[1+12*x])/(2*(4-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PARI

a(n) = if (n==0, 1, sum(m=0, n-1, (n-m)*binomial(n-1+m, m)*(-3)^m/n)); \\ Michel Marcus, Jul 30 2018

PARI

my(x='x+O('x^30)); Vec((7 +sqrt(1+12*x))/(2*(4-x))) \\ G. C. Greubel, May 03 2019

Sage

def a(n):
    if n == 0: return 1
    return hypergeometric([1-n, n], [-n], -3).simplify()
[a(n) for n in range(24)] # Peter Luschny, Nov 30 2014

Sage

((7 +sqrt(1+12*x))/(2*(4-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)*(-3)^m/n.

a(n) = (1/4)^n*(1 + 3*Sum_{k=0..n-1} C(k)*(-3*4)^k), n >= 1, a(0) = 1; with C(n) = A000108(n) (Catalan).

G.f.: (1+3*x*c(-3*x)/4)/(1-x/4) = 1/(1-x*c(-3*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = hypergeometric([1-n, n], [-n], -3) for n>0. - Peter Luschny, Nov 30 2014