A333306 a(n) = sqrt(Pi/4)*2^A048881(2*n)*L(2*n) where L(s) = lim_{t->s} (t/2)!/((1-t)/2)!.

1, 1, -1, 9, -45, 1575, -42525, 3274425, -42567525, 5746615875, -488462349375, 102088631019375, -6431583754220625, 1923043542511966875, -336532619939594203125, 136295711075535652265625, -3952575621190533915703125, 2083007352367411373575546875

Offset: 0

Author

Peter Luschny, May 17 2020

Keywords

• sign

Crossrefs

Cf. A048881, A048896, A046161, A034386, A278617.

Programs

Maple

L := s -> limit((factorial(t/2)/factorial((1-t)/2)), t=s):
G := n -> 2^(add(i, i = convert(n+1, base, 2)) - 1): # A048896
a := s -> sqrt(Pi/4)*G(2*s)*L(2*s): seq(a(n), n=0..17);

Mathematica

A333306[n_] := (-1)^n ((2 n)!/(1 - 2 n)) 2^(-2 n + DigitCount[2 n, 2, 1]);
Table[A333306[n], {n, 0, 17}]

Formula

a(n) = Z(2*n)*A048896(2*n)/2 where Z(n) = Pi^n*(n*Zeta(1 - n))/((1 - n)*Zeta(n)) for n >= 1.

a(n) = (-1)^n*(2*n)!/((1 - 2*n)*A046161(2*n)).

A034386(2*n-2)/2 divides a(n), i.e., all odd primes <= 2*(n-1) divide a(n).

The number of distinct prime divisors of a(n) is A278617(n).