OFFSET
0,2
COMMENTS
Hankel transform is A082761. - Paul Barry, Apr 14 2010
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
T.-X. He, L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, p 35.
FORMULA
a(n) = binomial(2*n, n)*Sum_{k=0..n} binomial(n, k)/binomial(2*k, k).
a(n) = 5*a(n-1) - 2*A000108(n-1).
G.f.: sqrt(1-4*x)/(1-5*x).
a(n) = (3*(3*n-2)/n)*a(n-1) - (10*(2*n-3)/n)*a(n-2), n >= 1, a(-1) := 0, a(0)=1 (homogeneous recursion).
a(n) = binomial(2*n,n)*hypergeom([ -n,1 ],[ 1/2 ],-1/4) (hypergeometric 2F1 form).
0 = a(n)*(+400*a(n+1) - 330*a(n+2) + 50*a(n+3)) + a(n+1)*(-30*a(n+1) + 71*a(n+2) - 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, May 25 2014
a(n) ~ 5^(n - 1/2). - Vaclav Kotesovec, Jul 07 2016
D-finite with recurrence n*a(n) +3*(-3*n+2)*a(n-1) +10*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jul 23 2017
EXAMPLE
G.f. = 1 + 3*x + 13*x^2 + 61*x^3 + 295*x^4 + 1447*x^5 + 7151*x^6 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Sqrt[ 1 - 4 x] / (1 - 5 x), {x, 0, n}]; (* Michael Somos, May 25 2014 *)
a[ n_] := Binomial[ 2 n, n] Hypergeometric2F1[ -n, 1, 1/2, -1/4]; (* Michael Somos, May 25 2014 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( 1 - 4*x + x * O(x^n)) / (1 - 5*x), n))}; /* Michael Somos, May 25 2014 */
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( Sqrt(1-4*x)/(1-5*x) )); // G. C. Greubel, Jul 28 2024
(SageMath)
def A046748_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( sqrt(1-4*x)/(1-5*x) ).list()
A046748_list(40) # G. C. Greubel, Jul 28 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang, Dec 11 1999
STATUS
approved