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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

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

Links

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

Crossrefs

Cf. A000108, A037964, A037972.

Programs

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

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 *)

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

Formula

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)