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

0,3

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

Formula

a(n) = (1/16)*(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)= 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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A000108, A037976.

Sequence in context: A259373 A076265 A114876 * A240626 A297916 A298541

Adjacent sequences: A037977 A037978 A037979 * A037981 A037982 A037983

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved