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

0,3

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, 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, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

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/(1-3x/(1-3x/(1-3x/(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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A215791 A175913 A239996 * A171991 A141371 A224079

Adjacent sequences: A064060 A064061 A064062 * A064064 A064065 A064066

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 13 2001

Status

approved