A064087 Generalized Catalan numbers C(4; n).

1, 1, 5, 41, 413, 4641, 55797, 702297, 9137549, 121909457, 1658755685, 22929591433, 321111942781, 4546112358529, 64958195967957, 935566629270201, 13567825195172973, 197957440018622769, 2903721563443327557, 42796201522669935081, 633443408407612143453

Offset

0,3

Comments

a(n+1) = Y_{n}(n+1) = Z_{n} in the Derrida et al. 1992 reference (see A064094) for alpha=4, beta=1 (or alpha=1, beta=4).

Links

Table of n, a(n) for n=0..20.

Formula

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

a(n) = (1/n)*Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(4^m) = ((-1/3)^n)*(1 - 4*Sum_{k=0..n-1} C(k)*(-12)^k), n >= 1, a(0) = 1, with C(n) = A000108(n) (Catalan).

a(n) = Sum_{k=0...n} A059365(n, k)*4^(n-k). - Philippe Deléham, Jan 19 2004

D-finite with recurrence: 3*n*a(n) + (-47*n+72)*a(n-1) + 8*(-2*n+3)*a(n-2) = 0. - R. J. Mathar, Jun 07 2013 [verified by Georg Fischer, Jul 06 2021]

a(n) = hypergeometric([1-n, n], [-n], 4) for n > 0. - Peter Luschny, Nov 30 2014

a(n) ~ 2^(4*n + 2) / (49*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

Mathematica

a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*4^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 09 2013 *)

Prog

(PARI)
a(n) = if(n<0, 0, polcoeff(serreverse((x-3*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */
(Sage)
def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], 4).simplify()
[a(n) for n in range(24)] # Peter Luschny, Nov 30 2014

Crossrefs

Cf. A064063 (C(3; n)).

Sequence in context: A378957 A199684 A177506 * A329123 A375437 A285064

Adjacent sequences: A064084 A064085 A064086 * A064088 A064089 A064090

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 13 2001

Status

approved