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 A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k). 14
 1, 35, 36465, 300540195, 79006629023595, 331884405207627584403, 22292910726608249789889125025, 11975573020964041433067793888190275875, 411646257111422564507234009694940786177843149765, 56592821660064550728377610673427602421565368547133335525825 (list; graph; refs; listen; history; text; internal format)
 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 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 Lim_{i->inf} a(i)*A285389(i+1)/(a(i+1)*A285389(i)) = 1. - Ralf Steiner, May 03 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 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)). Sequence in context: A271073 A271074 A249890 * A212926 A139473 A110596 Adjacent sequences: A285385 A285386 A285387 * A285389 A285390 A285391 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

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Last modified September 13 03:33 EDT 2024. Contains 375857 sequences. (Running on oeis4.)