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A181374 Let f(n) = Sum_{j>=1} j^n*3^j/binomial(2*j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives s_n.

%I A181374 #45 May 15 2020 16:23:55
%S A181374 3,18,156,1890,29496,563094,12709956,331109658,9777612432,
%T A181374 322738005150,11775245575836,470571509329506,20441566147934568,
%U A181374 959052902557542246,48330130399621041396,2603558645653906065834,149306059777139762896704,9081311859252750219451182,583927964165576868953730636
%N A181374 Let f(n) = Sum_{j>=1} j^n*3^j/binomial(2*j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives s_n.
%H A181374 Vaclav Kotesovec, <a href="/A181374/b181374.txt">Table of n, a(n) for n = 0..200</a>
%H A181374 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 A181374 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 A181374 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 A181374 a(n) ~ sqrt(2) * Pi * n^(n+1) / (3 * exp(n) * (log(4/3))^(n + 3/2)). - _Vaclav Kotesovec_, May 15 2020
%t A181374 f[n_] := Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}];
%t A181374 a[n_] := FindIntegerNullVector[{Pi/Sqrt[3], 1, N[-f[n], 20]}][[2]];
%t A181374 Table[s = a[n]; Print[s]; s, {n, 0, 8}] (* _Jean-François Alcover_, Sep 05 2018 *)
%t A181374 Table[Expand[FunctionExpand[FullSimplify[Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}]]]][[1]], {n, 0, 20}] (* _Vaclav Kotesovec_, May 14 2020 *)
%t A181374 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]]][[1]], {j, 0, 20}] (* _Vaclav Kotesovec_, May 15 2020 *)
%Y A181374 Cf. A185672 (r_n), A180875 and A014307 (2^j rather than 3^j).
%K A181374 nonn
%O A181374 0,1
%A A181374 _N. J. A. Sloane_, Feb 09 2011, following a suggestion from _Herb Conn_
%E A181374 More terms from _Vaclav Kotesovec_, May 14 2020