A036911 a(n) = (binomial(4*n, 2*n) + (-1)^n*binomial(2*n, n)^2)/2.

1, 1, 53, 262, 8885, 60626, 1778966, 14168988, 383358645, 3355615450, 85990654178, 803232328548, 19780031677718, 193873026294052, 4629016098160220, 47101568276955512, 1096960888092571317, 11503661742608944170, 262435310495071434602

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.

a(n) = (1/2)*((2*n+1)*A000108(2*n) + (-1)^n*(n+1)^2*A000108(n)^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

Cf. A000108, A037980.

Sequence in context: A142357 A142061 A107150 * A241488 A140851 A337428

Adjacent sequences: A036908 A036909 A036910 * A036912 A036913 A036914

Keyword

nonn

Author

N. J. A. Sloane

Status

approved