A037964 a(n) = (1/2)*(binomial(4*n, 2*n) - (-1)^n*binomial(2*n,n)).

0, 4, 32, 472, 6400, 92504, 1351616, 20060016, 300533760, 4537591960, 68923172032, 1052049834576, 16123800489472, 247959271674352, 3824345280321920, 59132290859989472, 916312070170755072

Offset

0,2

References

The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972; Formula (3.74), page 31.

Links

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

Formula

From R. J. Mathar, Feb 20 2015: (Start)

n*(2*n-1)*(n-1)*a(n) -12*(n-1)*(4*n^2-11*n+10)*a(n-1) +4*(38*n^3-333*n^2+715*n-435)*a(n-2) +48*(34*n^3-228*n^2+499*n-355)*a(n-3) +16*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4) = 0.

n*(n-1)*(2*n-1)*(5*n^2-15*n+11)*a(n) -4*(n-1)*(30*n^4-120*n^3 +161*n^2-82*n+12)*a(n-1) -4*(4*n-7)*(2*n-3)*(4*n-5)*(5*n^2-5*n+1)*a(n-2) = 0. (End)

From G. C. Greubel, Jun 20 2022: (Start)

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

a(n) = (1/2)*(A001448(n) - (-1)^n*A000984(n)).

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

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

Maple

A037964 := proc(n)
    binomial(4*n, 2*n)/2-(-1)^n*binomial(2*n, n)/2 ;
end proc: # R. J. Mathar, Feb 20 2015

Mathematica

With[{C= CatalanNumber}, Table[(1/2)*((2*n+1)*C[2*n] -(-1)^n*(n+1)*C[n]), {n, 0, 30}]] (* G. C. Greubel, Jun 20 2022 *)

Prog

(Magma) [(1/2)*((2*n+1)*Catalan(2*n) -(-1)^n*(n+1)*Catalan(n)): n in [0..30]]; // G. C. Greubel, Jun 20 2022
(SageMath) [sum(binomial(2*n, 2*k+1)^2 for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, Jun 20 2022

Crossrefs

Cf. A000108, A000984, A001448.

Sequence in context: A214379 A192500 A192486 * A089285 A209197 A086906

Adjacent sequences: A037961 A037962 A037963 * A037965 A037966 A037967

Keyword

nonn

Author

N. J. A. Sloane

Status

approved