OFFSET
0,3
REFERENCES
Identity (3.71) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 30.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..825
FORMULA
a(n) = (1/2)*(binomial(4*n, 2*n) + (-1)^n*binomial(2*n, n)^2).
From G. C. Greubel, Jun 22 2022: (Start)
a(n) = Sum_{k=0..n} binomial(2*n, 2*k)^2.
G.f.: (1/2)*( sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) + (2/Pi)*EllipticK(-16*x]) ). (End)
MATHEMATICA
Table[(Binomial[4n, 2n]+(-1)^n Binomial[2n, n]^2)/2, {n, 0, 20}] (* Harvey P. Dale, May 22 2013 *)
PROG
(Magma) [(1/2)*((2*n+1)*Catalan(2*n) + (-1)^n*(n+1)^2*Catalan(n)^2): n in [0..30]]; // G. C. Greubel, Jun 22 2022
(SageMath) b=binomial; [(1/2)*(b(4*n, 2*n) + (-1)^n*b(2*n, n)^2) for n in (0..30)] # G. C. Greubel, Jun 22 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved