A064063 Generalized Catalan numbers C(3; n).

1, 1, 4, 25, 190, 1606, 14506, 137089, 1338790, 13403950, 136846144, 1419257434, 14911016596, 158363649640, 1697452010230, 18338919413425, 199496184219910, 2183299541440150, 24021874198331080, 265559590979820910, 2948253066186839140, 32857382497018933060

Offset: 0

Author

Wolfdieter Lang, Sep 13 2001

Keywords

• nonn
• easy

Comments

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=3, beta =1 (or alpha=1, beta=3).

Hankel transform is A060722. - Paul Barry, Jan 30 2009

References

• S. Ramanujan, Modular Equations and Approximations to pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 39, equation (50).

Links

• Fung Lam, Table of n, a(n) for n = 0..925
• J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.
• Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
• S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Crossrefs

Cf. A064062 (C(2; n)).

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 6/(5+Sqrt(1-12*x)) )); // G. C. Greubel, May 02 2019

Mathematica

CoefficientList[Series[6/(5+Sqrt[1-12 x]),{x,0,50}],x]  (* Harvey P. Dale, Mar 11 2011 *)

PARI

a(n)=if(n<0,0,polcoeff(serreverse((x-2*x^2)/(1+x)^2+O(x^(n+1))),n)) \\ Ralf Stephan, Jun 12 2004

PARI

{a(n)= if(n<1, n==0, polcoeff( serreverse( (x-2*x^2)/ (1+x)^2 +x*O(x^n)), n))} /* Michael Somos, Apr 11 2007 */

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

Formula

G.f.: (1+3*x*c(3*x)/2)/(1+x/2) = 1/(1-x*c(3*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(3^m)/n.

a(n) = (-1/2)^n * (1 - 3*Sum_{k=0..n-1} C(k)*(-6)^k), n >= 1, a(0) = 1, with C(n) = A000108(n) (Catalan).

a(n) = Sum_{k=0..n} A059365(n, k)*3^(n-k). - Philippe Deléham, Jan 19 2004

Given the semi-axes a,b of an ellipse, then Ramanujan gave the highly accurate formula for the perimeter p = Pi((a+b) + (3(a-b)^2)/(10(a+b) + sqrt(a^2 + 14ab + b^2))). If we let h = ((a-b)/(a+b))^2, then (p/(Pi(a+b))-1)/4 = (3/4)* h/(10 + sqrt(4 - 3*h)) = 1*(h/16) + 1*(h/16)^2 + 4*(h/16)^3 + 25*(h/16)^4 + ... . - Michael Somos, Apr 11 2007

G.f.: 1/(1-x/(1-3*x/(1-3*x/(1-3*x/(1-.... = 1/(1-x-3*x^2/(1-6*x-9*x^2/(1-6*x-9*x^2/(1-.... (continued fractions). - Paul Barry, Jan 30 2009

G.f.: 6/(5+sqrt(1-12*x)). - Harvey P. Dale, Mar 11 2011

From Gary W. Adamson, Jul 12 2011: (Start)

a(n) = upper left term in M^n, M = the infinite square production matrix:

1, 1, 0, 0, 0, 0, ...

3, 3, 3, 0, 0, 0, ...

3, 3, 3, 3, 0, 0, ...

3, 3, 3, 3, 3, 0, ...

3, 3, 3, 3, 3, 3, ...

... (End)

D-finite with recurrence: 2*n*a(n) + (-23*n+36)*a(n-1) + 6*(-2*n+3)*a(n-2) = 0. - R. J. Mathar, Dec 03 2012 (Formula verified and used for computations. - Fung Lam, Mar 05 2014)

a(n) ~ 3^(n+1) * 4^n / (25*n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Mar 05 2014

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