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%I #34 May 15 2020 16:23:38
%S 4,20,172,2084,32524,620900,14014732,365100644,10781360524,
%T 355869575780,12984066273292,518879340911204,22540052170064524,
%U 1057507154836226660,53291594817628483852,2870834224548449841764,164633490033421041392524,10013579272685278891133540,643872718978606529940390412
%N Let f(n) = Sum_{j>=1} j^n*3^j/binomial(2*j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives r_n.
%H Vaclav Kotesovec, <a href="/A185672/b185672.txt">Table of n, a(n) for n = 0..200</a>
%H 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 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 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 a(n) ~ 2^(3/2) * n^(n+1) / (sqrt(3) * exp(n) * (log(4/3))^(n + 3/2)). - _Vaclav Kotesovec_, May 15 2020
%t f[n_] := Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}];
%t a[n_] := FindIntegerNullVector[{Pi/Sqrt[3], 1, N[-f[n], 20]}][[1]];
%t Table[r = a[n]; Print[r]; r, {n, 0, 8}] (* _Jean-François Alcover_, Sep 05 2018 *)
%t Table[Expand[FunctionExpand[FullSimplify[Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}]]]][[2]] * Sqrt[3]/Pi, {n, 0, 20}] (* _Vaclav Kotesovec_, May 14 2020 *)
%t S[k_, z_] := Sum[n!*(z/(4 - z))^n* StirlingS2[k + 1, n]*(1/n + Sum[(-1)^p*Pochhammer[1/2, p]/(p + 1)!* Binomial[n - 1, p]*(4/z)^(p + 1)*(Sqrt[z/(4 - z)]*ArcSin[Sqrt[z]/2] - 1/2*Sum[Gamma[l]/Pochhammer[1/2, l]*(z/4)^l, {l, 1, p}]), {p, 0, n - 1}]), {n, 1, k + 2}]; Table[Expand[Simplify[S[j, 3]]][[2]]*Sqrt[3]/Pi, {j, 0, 20}] (* _Vaclav Kotesovec_, May 15 2020 *)
%Y Cf. A181374 (s_n), A180875 and A014307 (2^j rather than 3^j).
%K nonn
%O 0,1
%A _N. J. A. Sloane_, Feb 09 2011, following a suggestion from _Herb Conn_
%E More terms from _Vaclav Kotesovec_, May 14 2020
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