A037980 - OEIS

%I #14 Aug 07 2024 09:21:15

%S 0,1,4,109,800,19501,168952,3979830,37566720,862687045,8615396504,

%T 193710517650,2015475061184,44516469004294,478043160040240,

%U 10399216983867484,114539008771344384,2459029841101222485,27657033766735102744,586949749681986718650,6719200545824895620800,141147097812860184921810

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

%D 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.)

%H G. C. Greubel, <a href="/A037980/b037980.txt">Table of n, a(n) for n = 0..825</a>

%F From _G. C. Greubel_, Jun 22 2022: (Start)

%F a(n)= A037976(n)/4.

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

%F 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 ).

%F 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)

%p A037980 := proc(n)

%p     binomial(4*n,2*n) -(-1)^n*binomial(2*n,n)+(1-(-1)^n)*binomial(2*n,n)^2 ;

%p     %/16 ;

%p end proc: # _R. J. Mathar_, Oct 20 2015

%t 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 *)

%o (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

%o (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

%Y Cf. A000108, A037976.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E More terms added by _G. C. Greubel_, Jun 22 2022