A270530 a(n) = Sum_{k=0..n}((binomial(2*k,k)/(k+1)*binomial(2*n+2,n-k))).
1, 5, 23, 105, 484, 2267, 10821, 52705, 262010, 1328768, 6867266, 36115455, 192954358, 1045481465, 5735154907, 31802349105, 178010615678, 1004542994462, 5709066033900, 32646940202200, 187701954810320
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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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
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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 *) -
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);
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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
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
Comments