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A185672 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. 2
4, 20, 172, 2084, 32524, 620900, 14014732, 365100644, 10781360524, 355869575780, 12984066273292, 518879340911204, 22540052170064524, 1057507154836226660, 53291594817628483852, 2870834224548449841764, 164633490033421041392524, 10013579272685278891133540, 643872718978606529940390412 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

FORMULA

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

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]}][[1]];

Table[r = a[n]; Print[r]; r, {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}]]]][[2]] * Sqrt[3]/Pi, {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]]][[2]]*Sqrt[3]/Pi, {j, 0, 20}] (* Vaclav Kotesovec, May 15 2020 *)

CROSSREFS

Cf. A181374 (s_n), A180875 and A014307 (2^j rather than 3^j).

Sequence in context: A032333 A357666 A068965 * A210438 A054474 A213144

Adjacent sequences: A185669 A185670 A185671 * A185673 A185674 A185675

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

More terms from Vaclav Kotesovec, May 14 2020

STATUS

approved

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Last modified December 9 11:21 EST 2022. Contains 358700 sequences. (Running on oeis4.)