A064089 Generalized Catalan numbers C(6; n).

1, 1, 7, 85, 1279, 21517, 387607, 7312789, 142648495, 2853691357, 58226571271, 1207062556261, 25351452769567, 538285926177325, 11535690316148215, 249189167966657845, 5420206822556721295

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=6, beta =1 (or alpha=1, beta=6).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..725

Crossrefs

Cf. A064088 (C(5, n)).

Cf. A000108, A059365.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (11 -Sqrt(1-24*x))/(2*(x+5)) )); // G. C. Greubel, May 02 2019

Mathematica

CoefficientList[Series[(11 -Sqrt[1-24*x])/(2*(x+5)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

PARI

a(n)=if(n<0,0,polcoeff(serreverse((x-5*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

PARI

my(x='x+O('x^20)); Vec((11 -sqrt(1-24*x))/(2*(x+5))) \\ G. C. Greubel, May 02 2019

Sage

((11 -sqrt(1-24*x))/(2*(x+5))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

Formula

G.f.: (1 + 6*x*c(6*x)/5)/(1+x/5) = 1/(1 - x*c(6*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(6^m)/n.

a(n) = (-1/5)^n*(1 - 6*Sum_{k=0..n-1} C(k)*(-30)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

a(n) = Sum_{k=0..n} A059365(n, k)*6^(n-k). - Philippe Deléham, Jan 19 2004

Conjecture: 5*n*a(n) +(-119*n+180)*a(n-1) +12*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

a(n) ~ 2^(3*n + 1) * 3^(n+1) / (121*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019