A064090 Generalized Catalan numbers C(7; n).

1, 1, 8, 113, 1982, 38886, 817062, 17981769, 409186310, 9549411950, 227307541448, 5497312072330, 134696099554276, 3336563455537768, 83419226227330722, 2102274863070771033, 53347639317495439302

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=7, beta =1 (or alpha=1, beta=7).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..690

Crossrefs

Cf. A064089 (C(6, n)).

Cf. A000108, A059365.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (13-Sqrt(1-28*x))/(2*(x+6)) )); // G. C. Greubel, May 02 2019

Mathematica

a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n-1+m, m]*7^m/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 09 2013 *)
CoefficientList[Series[(13 -Sqrt[1-28*x])/(2*(x+6)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

PARI

a(n)=if(n<0,0,polcoeff(serreverse((x-6*x^2)/(1+x)^2 +O(x^(n+1))), n)) /* Ralf Stephan */

PARI

my(x='x+O('x^20)); Vec((13-sqrt(1-28*x))/(2*(x+6))) \\ G. C. Greubel, May 02 2019

Sage

((13-sqrt(1-28*x))/(2*(x+6))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

Formula

G.f.: (1+7*x*c(7*x)/6)/(1+x/6) = 1/(1-x*c(7*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(7^m)/n.

a(n) = (-1/6)^n*(1 - 7*Sum_{k=0..n-1} C(k)*(-42)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

a(n) = Sum_{k=0..n} A059365(n, k)*7^(n-k). - Philippe Deléham, Jan 19 2004

Conjecture: 6*n*a(n) +(-167*n+252)*a(n-1) +14*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

a(n) ~ 4^n * 7^(n+1) / (169*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019