A285389 Denominator of Sum_{k=0..n^2-1} (-1)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k))/n.

1, 32, 32768, 268435456, 70368744177664, 295147905179352825856, 19807040628566084398385987584, 10633823966279326983230456482242756608, 365375409332725729550921208179070754913983135744, 50216813883093446110686315385661331328818843555712276103168

Offset: 1

Author

Ralf Steiner, Apr 18 2017

Keywords

• nonn
• frac

Comments

All terms are powers of 2.

Lim_{n->inf} A285388(n)/a(n) = 2/sqrt(Pi).

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..40

Crossrefs

Cf. A000079 (powers of 2), A285388 (numerators), A285406 (log base 2; A281264 + A007814), A190732 (2/sqrt(Pi)).

Cf. A000120, A007814, A285406.

Programs

Magma

[Denominator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021

Mathematica

Table[Denominator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n],{n,1,10}]
Denominator[Table[2^(1-2 n^2) n Binomial[2 n^2, n^2], {n, 1, 10}]] (* Ralf Steiner, Apr 22 2017 *)

Python

from sympy import binomial, Integer
def a(n): return (Integer(2)**(1-2*n**2) * Integer(n) * binomial(2*n**2, n**2)).denominator # Indranil Ghosh, Apr 27 2017

Sage

def A285389(n): return 2^(2*n^2 - 1 - (n^2).popcount() - valuation(n, 2))
[A285389(n) for n in (1..20)] # G. C. Greubel, Dec 12 2021

Formula

a(n) is the denominator of Sum_{k=0..n^2-1} (binomial(2k,k)/4^k)/n.

a(n) = A000079(A285406(n)).

a(n) = denominator of n*binomial(2*n^2, n^2)/2^(2*n^2 -1). - Ralf Steiner, Apr 22 2017