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A064325 Generalized Catalan numbers C(-3; n). 6
1, 1, -2, 13, -98, 826, -7448, 70309, -686090, 6865150, -70057772, 726325810, -7628741204, 81002393668, -868066319108, 9376806129493, -101988620430938, 1116026661667318, -12277755319108748, 135715825209716038, -1506587474535945788, 16789107646422189868, -187747069029477151328 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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..22.

FORMULA

a(n) = sum((n-m)*binomial(n-1+m, m)*((-3)^m)/n, m=0..n-1) = ((1/4)^n)*(1+3*sum(C(k)*(-3*4)^k, k=0..n-1)), 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

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-Fran├žois Alcover, Jul 30 2018 *)

PROG

(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

(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

CROSSREFS

Cf. A064334, A000108.

Sequence in context: A074614 A184019 A300633 * A123619 A187746 A030519

Adjacent sequences:  A064322 A064323 A064324 * A064326 A064327 A064328

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Sep 21 2001

STATUS

approved

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Last modified October 16 20:34 EDT 2018. Contains 316275 sequences. (Running on oeis4.)