OFFSET
1,2
COMMENTS
Editorial comment: This sequence arose from Ralf Steiner's attempt to prove Legendre's conjecture that there is a prime between N^2 and (N+1)^2 for all N. - N. J. A. Sloane, May 01 2017
LINKS
Indranil Ghosh, Table of n, a(n) for n = 1..40
FORMULA
a(n) is numerator of n*binomial(2 n^2, n^2)/2^(2*n^2 - 1). - Ralf Steiner, Apr 26 2017
MATHEMATICA
Table[Numerator[Sum[Binomial[2k, k]/4^k, {k, 0, n^2-1}]/n], {n, 1, 10}]
Numerator[Table[2^(1-2 n^2) n Binomial[2 n^2, n^2], {n, 1, 10}]] (* Ralf Steiner, Apr 22 2017 *)
PROG
(PARI) A285388(n) = numerator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)); \\ Antti Karttunen, Apr 27 2017
(PARI) a(n) = m=n*binomial(2*n^2, n^2); m>>valuation(m, 2) \\ David A. Corneth, Apr 27 2017
(Python)
from sympy import binomial, Integer
def a(n): return (Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).numerator() # Indranil Ghosh, Apr 27 2017
(Magma) [Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021
(Sage) [numerator( n*(n^2+1)*catalan_number(n^2)/2^(2*n^2-1) ) for n in (1..20)] # G. C. Greubel, Dec 11 2021
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Ralf Steiner, Apr 18 2017
EXTENSIONS
Edited (including the removal of the author's claim that this leads to a proof of the Legendre conjecture) by N. J. A. Sloane, May 01 2017
Formula section edited by M. F. Hasler, May 02 2017
Edited by N. J. A. Sloane, May 10 2017
STATUS
approved