login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A064087 Generalized Catalan numbers C(4; n). 5
1, 1, 5, 41, 413, 4641, 55797, 702297, 9137549, 121909457, 1658755685, 22929591433, 321111942781, 4546112358529, 64958195967957, 935566629270201, 13567825195172973, 197957440018622769, 2903721563443327557, 42796201522669935081, 633443408407612143453 (list; graph; refs; listen; history; text; internal format)
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) = sum((n-m)*binomial(n-1+m, m)*(4^m)/n, m=0..n-1) = ((-1/3)^n)*(1-4*sum(C(k)*(-12)^k, k=0..n-1)), 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

Conjecture: 3*n*a(n) +(-47*n+72)*a(n-1) +8*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

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

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: A058475 A199684 A177506 * A285064 A232685 A081215

Adjacent sequences:  A064084 A064085 A064086 * A064088 A064089 A064090

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 13 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 16 12:51 EDT 2018. Contains 316263 sequences. (Running on oeis4.)