A037976 a(n) = (1/4)*(binomial(4*n, 2*n) - (-1)^n*binomial(2*n, n) + (1-(-1)^n)*binomial(2*n, n)^2).

0, 4, 16, 436, 3200, 78004, 675808, 15919320, 150266880, 3450748180, 34461586016, 774842070600, 8061900244736, 178065876017176, 1912172640160960, 41596867935469936, 458156035085377536

Offset

0,2

References

The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972. (See (3.75) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 31.)

Links

G. C. Greubel, Table of n, a(n) for n = 0..825

Formula

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

From G. C. Greubel, Jun 22 2022: (Start)

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(2*n, 2*k+1)^2.

a(n) = (1/4)*( (2*n+1)*A000108(2*n) - (-1)^n*(n+1)*A000108(n) + (1-(-1)^n*(n+1)^2*A000108(n)^2 ).

G.f.: (1/4)*(sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) - 1/sqrt(1+4*x)) + (1/(2*Pi))*( EllipticK(16*x) - EllipticK(-16*x)). (End)

Maple

A037976 := proc(n)
    binomial(4*n, 2*n)/4-(-1)^n*binomial(2*n, n)/4+(1-(-1)^n)*binomial(2*n, n)^2/4 ;
end proc:
seq(A037976(n), n=0..30) ; # R. J. Mathar, Jul 26 2015

Mathematica

With[{B=Binomial}, Table[(1/4)*(B[4*n, 2*n] +B[2*n, n]^2 -2*(-1)^n*B[B[2*n, n] +1, 2]), {n, 0, 30}]] (* G. C. Greubel, Jun 22 2022 *)

Prog

(Magma) [(1/4)*((2*n+1)*Catalan(2*n) -(-1)^n*(n+1)*Catalan(n) +(1-(-1)^n)*(n+1)^2*Catalan(n)^2): n in [0..30]]; // G. C. Greubel, Jun 22 2022
(SageMath) b=binomial; [(1/4)*(b(4*n, 2*n) -(-1)^n*b(2*n, n) +(1-(-1)^n)*b(2*n, n)^2) for n in (0..30)] # G. C. Greubel, Jun 22 2022

Crossrefs

Cf. A000108, A037964, A037972.

Sequence in context: A156337 A000874 A061580 * A013138 A262546 A013006

Adjacent sequences: A037973 A037974 A037975 * A037977 A037978 A037979

Keyword

nonn

Author

N. J. A. Sloane

Status

approved