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%I #20 Apr 18 2018 04:51:41
%S 4,22,218,3048,54852,1206696,31373856,941214240,32001274080,
%T 1216048334400,51074029319040,2349405341418240,117470266991078400,
%U 6343394416560230400,367916876148039321600,22810846321004081356800,1505515857183654020812800,105386110002813935877120000
%N a(n) = 2*((2*n-1)*a(n-1) - (n-2)!), with a(1) = 4, n > 1.
%H Travis Sherman, <a href="http://math.arizona.edu/~rta/001/sherman.travis/series.pdf">Summation of Glaisher- and Apery-like Series</a>, University of Arizona, May 23 2000, p. 14, (3.79) - (3.83).
%F a(n) = f1(n)*9*(n-1)!, where f1(n) corresponds to the x values such that Sum_{k>=0} 1/(binomial(2*k,k)*(k+n))) = x*Pi*sqrt(3) - y*Pi^2 - z. (See examples for connection with a(n) in terms of material at Links section.)
%F f2(n) corresponds to the y values, so f2(n) = (1/9)*((2*n-1)!/((n-1)!)^2).
%F a(n) ~ Pi * 2^(2*n + 1/2) * n^n / (sqrt(3) * exp(n)). - _Vaclav Kotesovec_, Apr 13 2018
%e Examples ((3.79) - (3.83)) at page 14 in Links section as follows, respectively.
%e For n=1, f1(1) = 4/9, so a(1) = 4.
%e For n=2, f1(2) = 22/9, so a(2) = 22.
%e For n=3, f1(3) = 109/9, so a(3) = 218.
%e For n=4, f1(4) = 508/9, so a(4) = 3048.
%e For n=5, f1(5) = 4571/18, so a(5) = 54852.
%t RecurrenceTable[{a[n] == 2*((2*n - 1)*a[n-1] - (n-2)!), a[1] == 4}, a, {n, 1, 20}] (* _Vaclav Kotesovec_, Apr 13 2018 *)
%t Table[FullSimplify[4^n*Sqrt[Pi/3] * Gamma[n + 1/2] + Gamma[n] * Hypergeometric2F1[1, n, n + 3/2, 1/4]/(2*n + 1)], {n, 1, 20}] (* _Vaclav Kotesovec_, Apr 13 2018 *)
%t nmax = 15; Table[CoefficientList[Expand[FunctionExpand[ Table[Sum[1/(Binomial[2*j, j]*(j + m)), {j, 0, Infinity}], {m, 1, nmax}]]], Pi][[n, 2]]*9*(n-1)! / Sqrt[3], {n, 1, nmax}] (* _Vaclav Kotesovec_, Apr 13 2018 *)
%Y Cf. A302770.
%K nonn
%O 1,1
%A _Detlef Meya_, Apr 13 2018
%E More terms from _Vaclav Kotesovec_, Apr 13 2018
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