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
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.
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms added by G. C. Greubel, Jun 22 2022
STATUS
approved