A064093 Generalized Catalan numbers C(10; n).

1, 1, 11, 221, 5531, 154941, 4649451, 146150061, 4750427771, 158361063581, 5384626548491, 186023930383501, 6511108452179611, 230400987949757821, 8228844334672249131, 296245683962814194541, 10739133812893020645051

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=10, beta =1 (or alpha=1, beta=10).

In general, for m>=1, C(m; n) ~ m * (4*m)^n / ((2*m - 1)^2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 10 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..620

Crossrefs

Cf. A064092 (C(9, n)).

Cf. A000108, A059365.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (19 - Sqrt(1-40*x))/(2*(x+9)) )); // G. C. Greubel, May 02 2019

Mathematica

CoefficientList[Series[(19 -Sqrt[1-40*x])/(2*(x+9)), {x, 0, 20}], x] (* G. C. Greubel, May 02 2019 *)

PARI

my(x='x+O('x^20)); Vec((19 -sqrt(1-40*x))/(2*(x+9))) \\ G. C. Greubel, May 02 2019

Sage

((19 -sqrt(1-40*x))/(2*(x+9))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019

Formula

G.f.: (1 + 10*x*c(10*x)/9)/(1+x/9) = 1/(1 - x*c(10*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(10^m)/n.

a(n) = (-1/9)^n*(1 - 10*Sum_{k=0..n-1} C(k)*(-90)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

a(n) = Sum_{k=0..n} A059365(n, k)*10^(n-k). - Philippe Deléham, Jan 19 2004

a(n) ~ 2^(3*n + 1) * 5^(n+1) / (361*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 10 2019