A064092 Generalized Catalan numbers C(9; n).

1, 1, 10, 181, 4078, 102826, 2777212, 78571837, 2298558934, 68964092542, 2110472708140, 65620725560578, 2067160250751436, 65833929303952564, 2116166898185821792, 68565914052628406221, 2237022199842087256678

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=9, beta =1 (or alpha=1, beta=9).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..640

Crossrefs

Cf. A064091 (C(8, n)).

Cf. A000108, A059365.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (17 - Sqrt(1-36*x))/(2*(x+8)) )); // G. C. Greubel, May 02 2019

Mathematica

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n-1+m, m]*9^m/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(17 -Sqrt[1-36*x])/(2*(x+8)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

PARI

a(n)=if(n<0,0,polcoeff(serreverse((x-8*x^2)/(1+x)^2+O(x^(n+1))), n)) /* Ralf Stephan */

PARI

my(x='x+O('x^20)); Vec((17 -sqrt(1-36*x))/(2*(x+8))) \\ G. C. Greubel, May 02 2019

Sage

((17 -sqrt(1-36*x))/(2*(x+8))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

Formula

G.f.: (1 + 9*x*c(9*x)/8)/(1+x/8) = 1/(1 - x*c(9*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(9^m)/n.

a(n) = (-1/8)^n*(1 - 9*Sum_{k=0..n-1} C(k)*(-72)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

a(n) = Sum_{k=0..n} A059365(n, k)*9^(n-k). - Philippe Deléham, Jan 19 2004

Conjecture: 8*n*a(n) +(-287*n+432)*a(n-1) +18*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

a(n) ~ 4^n * 9^(n+1) / (289*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019