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, 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

Offset: 0

Author

N. J. A. Sloane, Feb 09 2011, following a suggestion from Herb Conn

Keywords

• nonn

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.

Crossrefs

Cf. A098830 (s_n), A181334 (r_n), A181374, A180875, A014307.

Programs

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 *)

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

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

Extensions

a(11)-a(57) from Nathaniel Johnston, Apr 07 2011