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 A185585 Let f(n) = Sum_{j>=1} j^n/binomial(2*j,j) = r_n*Pi*sqrt(3)/3^{t_n} + s_n/3; sequence gives t_n. 3
 3, 3, 4, 5, 5, 5, 6, 5, 7, 8, 8, 9, 10, 10, 10, 10, 8, 11, 12, 12, 13, 14, 14, 13, 15, 13, 16, 17, 17, 18, 19, 19, 19, 20, 19, 21, 22, 22, 23, 24, 24, 24, 24, 23, 24, 25, 25, 26, 27, 27, 26, 28, 26, 29, 30, 30, 31, 32 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS Petros Hadjicostas, Table of n, a(n) for n = 0..300 F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011. F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130. D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457. FORMULA a(n) = ilog[3](denominator(2*Sum_{m=1..n+1} Sum_{p=0..m-1} (-1)^p * (m!/((p+1)*3^(m+2))) * Stirling2(n+1,m) * binomial(2*p,p) * binomial(m-1,p))), where ilog[3](3^k) = k. [It follows from Theorem 1 in Dyson et al. (2013).] - Petros Hadjicostas, May 14 2020 MAPLE # The function LehmerSer is defined in A181334. a := n -> ilog[3](denom(LehmerSer(n))): seq(a(n), n=0..57); # Peter Luschny, May 15 2020 MATHEMATICA f[n_] := Sum[j^n/Binomial[2*j, j], {j, 1, Infinity}]; a[n_] := 1 + Log[3, Denominator[Expand[FunctionExpand[f[n]]][[2, 1]]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 60}] (* Jean-François Alcover, Nov 24 2017 *) PROG (PARI) a(n) = logint(denominator(2*sum(m=1, n+1, sum(p=0, m-1, (-1)^p*(m!/((p+1)*3^(m+2)))*stirling(n+1, m, 2)*binomial(2*p, p)*binomial(m-1, p)))), 3) \\ Petros Hadjicostas, May 14 2020 CROSSREFS Cf. A098830 (s_n), A181334 (r_n), A181374, A180875, A014307. Sequence in context: A064631 A298200 A072648 * A072945 A307912 A274004 Adjacent sequences:  A185582 A185583 A185584 * A185586 A185587 A185588 KEYWORD nonn AUTHOR N. J. A. Sloane, Feb 09 2011, following a suggestion from Herb Conn EXTENSIONS a(11)-a(57) from Nathaniel Johnston, Apr 07 2011 STATUS approved

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Last modified June 24 02:56 EDT 2021. Contains 345415 sequences. (Running on oeis4.)