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%I #192 Sep 08 2022 08:46:19
%S 1,35,36465,300540195,79006629023595,331884405207627584403,
%T 22292910726608249789889125025,11975573020964041433067793888190275875,
%U 411646257111422564507234009694940786177843149765,56592821660064550728377610673427602421565368547133335525825
%N a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).
%C 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
%H Indranil Ghosh, <a href="/A285388/b285388.txt">Table of n, a(n) for n = 1..40</a>
%F a(n) is numerator of n*binomial(2 n^2, n^2)/2^(2*n^2 - 1). - _Ralf Steiner_, Apr 26 2017
%F a(n) = numerator(n*A201555(n) / (A060757(n)/2)) = n*A201555(n) / 2^(A285717(n)) = A000265(n*A201555(n)). [Using _Ralf Steiner_'s formula and A285717(n) <= A056220(n), cf. A285406.] - _Antti Karttunen_, Apr 27 2017
%F Lim_{i->inf} a(i)*A285389(i+1)/(a(i+1)*A285389(i)) = 1. - _Ralf Steiner_, May 03 2017
%t Table[Numerator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n],{n,1,10}]
%t Numerator[Table[2^(1-2 n^2) n Binomial[2 n^2,n^2],{n,1,10}]] (* _Ralf Steiner_, Apr 22 2017 *)
%o (PARI) A285388(n) = numerator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)); \\ _Antti Karttunen_, Apr 27 2017
%o (PARI) a(n) = m=n*binomial(2*n^2, n^2);m>>valuation(m,2) \\ _David A. Corneth_, Apr 27 2017
%o (Python)
%o from sympy import binomial, Integer
%o def a(n): return (Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).numerator() # _Indranil Ghosh_, Apr 27 2017
%o (Magma) [Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // _G. C. Greubel_, Dec 11 2021
%o (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
%Y Cf. A000079, A000265, A056220, A060757, A201555, A285389 (denominators), A285406, A280655 (similar), A190732 (2/sqrt(Pi)), A285738 (greatest prime factor), A285717, A285730, A285786, A286264, A000290 (n^2), A056220 (2*n^2 -1), A286127 (sum a(n-1)/a(n)).
%K nonn,frac
%O 1,2
%A _Ralf Steiner_, Apr 18 2017
%E 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
%E Formula section edited by _M. F. Hasler_, May 02 2017
%E Edited by _N. J. A. Sloane_, May 10 2017
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