A285406 Base-2 logarithm of denominator of Sum_{k=0..n^2-1} (-1)^k*sqrt(Pi)/(Gamma(1/2-k)*Gamma(1+k)*n).

0, 5, 15, 28, 46, 68, 94, 123, 158, 195, 236, 283, 333, 387, 445, 506, 574, 643, 716, 794, 875, 961, 1054, 1146, 1244, 1346, 1451, 1562, 1676, 1794, 1916, 2041, 2174, 2307, 2444, 2586, 2731, 2881, 3034, 3193, 3356, 3520, 3690, 3864, 4041, 4227, 4413, 4601, 4796, 4993

Offset: 1

Author

Ralf Steiner, Apr 18 2017

Keywords

• nonn

Comments

Needed for studying of Wallis-kind products of central binomials.

Crossrefs

Cf. A000120, A000523, A000984, A007814, A011371, A056220, A201555, A285389, A281264, A285717.

Programs

Mathematica

Log[2,Table[Denominator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n], {n,1,50}]]
Log[2,Denominator[Table[2^(1-2 n^2) n Binomial[2 n^2,n^2],{n,1,50}]]] (* Ralf Steiner, Apr 22 2017 *)

PARI

a(n) = logint(denominator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)), 2); \\ Indranil Ghosh, Apr 27 2017

PARI

val(n, p) = my(r=0); while(n, r+=n\=p);r
a(n) = 2*n^2-1 - valuation(n, 2) - val(2*n^2, 2) + 2*val(n^2, 2) \\ David A. Corneth, Apr 28 2017

Python

from sympy import binomial, integer_log, Integer
def a(n): return integer_log((Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).denominator, 2)[0] # Indranil Ghosh, Apr 27 2017

Scheme

(define (A285406 n) (- (* 2 n n) (A007814 n) (A000120 (* n n)) 1)) ;; Antti Karttunen, Apr 28 2017

Formula

a(n) = A000523(A285389(n)).

a(n) = A056220(n) - A285717(n) = (2*(n^2)) - A007814(n) - A000120(n^2) - 1. - Antti Karttunen, Apr 28 2017, based on Vladimir Shevelev's Jul 20 2009 formula in A000984