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.

3, 18, 156, 1890, 29496, 563094, 12709956, 331109658, 9777612432, 322738005150, 11775245575836, 470571509329506, 20441566147934568, 959052902557542246, 48330130399621041396, 2603558645653906065834, 149306059777139762896704, 9081311859252750219451182, 583927964165576868953730636

Offset: 0

Author

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

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200
• 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. A185672 (r_n), A180875 and A014307 (2^j rather than 3^j).

Programs

Mathematica

f[n_] := Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}];
a[n_] := FindIntegerNullVector[{Pi/Sqrt[3], 1, N[-f[n], 20]}][[2]];
Table[s = a[n]; Print[s]; s, {n, 0, 8}] (* Jean-François Alcover, Sep 05 2018 *)
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 *)
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 *)

Formula

a(n) ~ sqrt(2) * Pi * n^(n+1) / (3 * exp(n) * (log(4/3))^(n + 3/2)). - Vaclav Kotesovec, May 15 2020

Extensions

More terms from Vaclav Kotesovec, May 14 2020