This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181334 #64 Feb 10 2025 04:42:21
%S A181334 2,2,10,74,238,938,13130,23594,1298462,26637166,201403930,5005052234,
%T A181334 135226271914,1315508114654,13747435592810,153590068548062,
%U A181334 202980764290906,69141791857625242,2766595825017102650,38897014541363246798,1724835471991750464238,80219728936311383557694
%N A181334 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 r_n.
%H A181334 Petros Hadjicostas, <a href="/A181334/b181334.txt">Table of n, a(n) for n = 0..101</a>
%H A181334 F. J. Dyson, N. E. Frankel and M. L. Glasser, <a href="http://arxiv.org/abs/1009.4274">Lehmer's Interesting Series</a>, arXiv:1009.4274 [math-ph], 2010-2011.
%H A181334 F. J. Dyson, N. E. Frankel and M. L. Glasser, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.02.116">Lehmer's interesting series</a>, Amer. Math. Monthly, 120 (2013), 116-130.
%H A181334 D. H. Lehmer, <a href="https://www.jstor.org/stable/2322496">Interesting series involving the central binomial coefficient</a>, Amer. Math. Monthly, 92(7) (1985), 449-457.
%F A181334 a(n) = numerator(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)). [It follows from Theorem 1 in Dyson et al. (2010-2011, 2013).] - _Petros Hadjicostas_, May 15 2020
%p A181334 LehmerSer := n -> 2*add(add((-1)^p*(m!/((p+1)*3^(m+2)))*Stirling2(n+1,m)
%p A181334 *binomial(2*p, p)*binomial(m-1, p), p=0..m-1), m=1..n+1):
%p A181334 a := n -> numer(LehmerSer(n)): seq(a(n), n=0..21);
%p A181334 # (after _Petros Hadjicostas_) _Peter Luschny_, May 15 2020
%t A181334 f[n_] := Sum[j^n/Binomial[2*j, j], {j, 1, Infinity}];
%t A181334 a[n_] := Expand[ FunctionExpand[ f[n] ] ][[2, 1]] // Numerator;
%t A181334 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Nov 24 2017 *)
%o A181334 (PARI) a(n)=numerator(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)))) \\ _Petros Hadjicostas_, May 15 2020
%Y A181334 Cf. A098830 (s_n), A185585 (t_n), A181374, A180875, A014307.
%K A181334 nonn
%O A181334 0,1
%A A181334 _N. J. A. Sloane_, Feb 09 2011, following a suggestion from _Herb Conn_
%E A181334 a(11)-a(21) from _Nathaniel Johnston_, Apr 07 2011