A064091 Generalized Catalan numbers C(8; n).

1, 1, 9, 145, 2905, 65121, 1563561, 39322929, 1022586105, 27272680705, 741894295369, 20504949587409, 574176887116441, 16254518495907745, 464436319229036265, 13376293681432402545, 387925710986712480825

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=8, beta =1 (or alpha=1, beta=8).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A064090 (C(7, n)).

Cf. A000108, A059365.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (15 - Sqrt(1-32*x))/(2*(x+7)) )); // G. C. Greubel, May 02 2019

Mathematica

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n+m-1, m]*(8^m)/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 21 2013 *)
Table[FullSimplify[(-1)^(2*n)*2^(3+5*n)*(1/2*(2*n-1))! Hypergeometric2F1[1,1/2+n,2+n,-224]/(Sqrt[Pi]*(n+1)!)],{n,0,20}] (* Vaclav Kotesovec, Aug 13 2013 *)
CoefficientList[Series[(15 -Sqrt[1-32*x])/(2*(x+7)), {x,0,20}], x] (* G. C. Greubel, May 02 2019 *)

PARI

a(n)=if(n<0,0,polcoeff(serreverse((x-7*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

PARI

my(x='x+O('x^20)); Vec((15 -sqrt(1-32*x))/(2*(x+7))) \\ G. C. Greubel, May 02 2019

Sage

((15 -sqrt(1-32*x))/(2*(x+7))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

Formula

G.f.: (1 + 8*x*c(8*x)/7)/(1+x/7) = 1/(1 - x*c(8*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(8^m)/n.

a(n) = (-1/7)^n*(1 - 8*Sum_{k=0..n-1} C(k)*(-56)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

a(n) = Sum_{k=0..n} A059365(n, k)*8^(n-k). - Philippe Deléham, Jan 19 2004

Conjecture: 7*n*a(n) +(-223*n+336)*a(n-1) +16*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

a(n) ~ 2^(5*n+3)/(225*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013