OFFSET
0,2
COMMENTS
Binomial transform of Catalan numbers.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1/(2*x*sqrt(1-4*x))+(-sqrt(((5*x+2*sqrt(1-4*x)-2))/(x^3*(4-16*x)))).
a(n) ~ 5^(2*n + 7/2) / (3^(3/2) * sqrt(Pi) * n^(3/2) * 2^(2*n+4)). - Vaclav Kotesovec, Mar 18 2016
Conjecture: 2*n*(2*n+3)*(n+1)*a(n) -n*(77*n^2+27*n-4)*a(n-1) +(549*n^3-987*n^2+686*n-168)*a(n-2) -20*(2*n-3)*(43*n^2-104*n+70)*a(n-3) +500*(2*n-5)*(n-2)*(2*n-3)*a(n-4)=0. - R. J. Mathar, Jun 07 2016
Conjecture: 2*n*(2*n+3)*(n+3)*(n+1)*a(n) -n*(57*n^3+228*n^2+107*n+8)*a(n-1) +4*(2*n-1) *(33*n^3+99*n^2-88*n+36)*a(n-2) -100*(n-1)*(2*n-1)*(2*n-3)*(n+4)*a(n-3)=0. - R. J. Mathar, Jun 07 2016
MAPLE
A270530 := proc(n)
add(binomial(2*k, k)/(k+1)*binomial(2*n+2, n-k), k=0..n) ;
end proc: # R. J. Mathar, Jun 07 2016
MATHEMATICA
CoefficientList[Series[1/(2*x*Sqrt[1 - 4*x]) + (-Sqrt[((5*x + 2*Sqrt[1 - 4*x] - 2))/(x^3*(4 - 16*x))]), {x, 0, 50}], x] (* G. C. Greubel, Apr 09 2017 *)
PROG
(Maxima)
a(n):=sum((binomial(2*k, k)/(k+1)*binomial(2*n+2, n-k)), k, 0, n);
makelist(coeff(taylor(1/(2*x*sqrt(1-4*x))+(-sqrt(((5*x+2*sqrt(1-4*x)-2))/(x^3*(4-16*x)))), x, 0, 10), x, n), n, 0, 10);
(PARI) x='x+O('x^50); Vec(1/(2*x*sqrt(1-4*x))+(-sqrt(((5*x+2*sqrt(1-4*x)-2))/(x^3*(4-16*x))))) \\ G. C. Greubel, Apr 09 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Mar 18 2016
STATUS
approved