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
Examples
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, ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Catalan Number
Programs
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Magma
[Numerator(Binomial(2*n,n)/Factorial(n+1)): n in [0..30]]; // G. C. Greubel, Jan 17 2019
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Maple
seq(numer(binomial(2*n,n)/(n+1)!),n=0..30); # Vladeta Jovovic, Dec 03 2008
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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 *) -
PARI
vector(30, n, n--; numerator(binomial(2*n,n)/(n+1)!)) \\ G. C. Greubel, Jan 17 2019
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Sage
[numerator(binomial(2*n,n)/factorial(n+1)) for n in (0..30)] # G. C. Greubel, Jan 17 2019
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)).