OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{j=0..(n+2)} C(n+2*j-1,j)*(-1)^(j+n)*C(2*n+2,j+n))/(n+1) -delta(n,0).
a(n) ~ (5+3*sqrt(5)) * 2^(2*n+1) / (5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 18 2014
Conjecture: 2*(2*n+1)*(n+2)*(n+1)*a(n) +(n+1)*(n^2-27*n+2)*a(n-1) +2*(-73*n^3+204*n^2-167*n+6)*a(n-2) +12*(n-3)*(2*n-3)*(4*n-7)*a(n-3) +216*(2*n-5)*(n-3)*(2*n-3)*a(n-4)=0. - R. J. Mathar, Apr 02 2014
MATHEMATICA
CoefficientList[Series[-16/(Sqrt[12*x+2*Sqrt[1-4*x]+2]-Sqrt[1-4*x] -1)^2+1/x^2-1, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 18 2014 *)
Flatten[{1, Table[Sum[Binomial[n+2*j-1, j+n-1]*(-1)^(j+n)*Binomial[2*n+2, j+n], {j, 0, n+2}]/(n+1), {n, 1, 20}]}] (* Vaclav Kotesovec, Mar 18 2014 *)
PROG
(Maxima)
a(n):=(sum(binomial(n+2*j-1, j)*(-1)^(j+n)*binomial(2*n+2, j+n), j, 0, n+2))/(n+1)-kron_delta(n, 0);
(PARI) x='x+O('x^50); Vec(-16/(sqrt(12*x+2*sqrt(1-4*x)+2)-sqrt(1-4*x) -1)^2 + 1/x^2 -1) \\ G. C. Greubel, Jun 01 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Mar 17 2014
STATUS
approved