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a(n) = (n!/floor(n/2)!^2)^2.
1

%I #9 May 11 2022 17:07:33

%S 1,1,4,36,36,900,400,19600,4900,396900,63504,7683984,853776,144288144,

%T 11778624,2650190400,165636900,47869064100,2363904400,853369488400,

%U 34134779536,15053437775376,497634306624,263248548204096,7312459672336,4570287295210000,108172480360000

%N a(n) = (n!/floor(n/2)!^2)^2.

%F a(n) = [x^n] (2/(Pi*(1 - 16*x^2)^2))*(2*x*EllipticE(4*x) + (16*x^2 - 1)*(16*x^2 - 1 + x)*EllipticK(4*x)).

%p seq((n!/iquo(n,2)!^2)^2, n = 0..26); # Or:

%p gf := (2/(Pi*(1 - 16*x^2)^2))*(2*x*EllipticE(4*x) + (16*x^2 - 1)*(16*x^2 - 1 + x)* EllipticK(4*x)): ser := series(gf, x, 30): seq(coeff(ser, x, n), n = 0..26);

%t Table[(n!/(Floor[n/2]!)^2)^2,{n,0,30}] (* _Harvey P. Dale_, May 11 2022 *)

%Y Central column of A328001.

%Y Even bisection is A002894.

%K nonn

%O 0,3

%A _Peter Luschny_, Oct 19 2019