A144186 Numerators of series expansion of the e.g.f. for the Catalan numbers.

1, 1, 1, 5, 7, 7, 11, 143, 143, 2431, 4199, 4199, 7429, 7429, 7429, 215441, 392863, 392863, 20677, 765049, 765049, 31367009, 58642669, 58642669, 2756205443, 2756205443, 2756205443, 146078888479, 5037203051, 5037203051, 9586934839

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Catalan Number

Formula

The e.g.f. is Sum_{n >= 0} (x^n/n!)*binomial(2n,n)/(n+1).

E.g.f.: exp(2*x)*(BesselI(0, 2*x) - BesselI(1, 2*x)).

Example

E.g.f. = 1 + x + x^2 + (5*x^3)/6 + (7*x^4)/12 + ...
The coefficients continue like this: 1, 1, 1, 5/6, 7/12, 7/20, 11/60, 143/1680, 143/4032, 2431/181440, 4199/907200, 4199/2851200, 7429/17107200, 7429/62270208, ...

Maple

seq(numer(binomial(2*n, n)/(n+1)!), n=0..30); # Vladeta Jovovic, Dec 03 2008

Mathematica

With[{m = 30}, CoefficientList[Series[E^(2*x)*(BesselI[0, 2*x] - BesselI[1, 2*x]), {x, 0, m}], x]]//Numerator (* G. C. Greubel, Jan 17 2019 *)

Prog

(PARI) vector(30, n, n--; numerator(binomial(2*n, n)/(n+1)!)) \\ G. C. Greubel, Jan 17 2019
(Magma) [Numerator(Binomial(2*n, n)/Factorial(n+1)): n in [0..30]]; // G. C. Greubel, Jan 17 2019
(Sage) [numerator(binomial(2*n, n)/factorial(n+1)) for n in (0..30)] # G. C. Greubel, Jan 17 2019

Crossrefs

Cf. A000108, A144187.

Sequence in context: A231935 A216835 A033932 * A246458 A153979 A126992

Adjacent sequences: A144183 A144184 A144185 * A144187 A144188 A144189

Keyword

nonn,frac

Author

Eric W. Weisstein, Sep 13 2008

Status

approved