A037980 a(n) = (1/16)*( binomial(4*n, 2*n) - (-1)^n*binomial(2*n, n) + (1-(-1)^n)*binomial(2*n, n)^2 ).

0, 1, 4, 109, 800, 19501, 168952, 3979830, 37566720, 862687045, 8615396504, 193710517650, 2015475061184, 44516469004294, 478043160040240, 10399216983867484, 114539008771344384, 2459029841101222485, 27657033766735102744, 586949749681986718650, 6719200545824895620800, 141147097812860184921810

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 Identity (3.75) divided by four in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 31.)

Links

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

Crossrefs

Cf. A000108, A037976.

Programs

Magma

[(1/16)*((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

A037980 := proc(n)
    binomial(4*n,2*n) -(-1)^n*binomial(2*n,n)+(1-(-1)^n)*binomial(2*n,n)^2 ;
    %/16 ;
end proc: # R. J. Mathar, Oct 20 2015

Mathematica

With[{B=Binomial}, Table[(1/16)*(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; [(1/16)*(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)= A037976(n)/4.

a(n) = (1/4)*Sum_{k=0..floor((n-1)/2)} binomial(2*n, 2*k+1)^2.

a(n) = (1/16)*( (2*n+1)*A000108(2*n) - (-1)^n*(n+1)*A000108(n) + (1-(-1)^n)*(n+1)^2*A000108(n)^2 ).

G.f.: (1/16)*(sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) - 1/sqrt(1+4*x)) + (1/(8*Pi))*( EllipticK(16*x) - EllipticK(-16*x)). (End)

Extensions

More terms added by G. C. Greubel, Jun 22 2022