A302769 a(n) = 2*((2*n-1)*a(n-1) - (n-2)!), with a(1) = 4, n > 1.

4, 22, 218, 3048, 54852, 1206696, 31373856, 941214240, 32001274080, 1216048334400, 51074029319040, 2349405341418240, 117470266991078400, 6343394416560230400, 367916876148039321600, 22810846321004081356800, 1505515857183654020812800, 105386110002813935877120000

Offset

1,1

Links

Table of n, a(n) for n=1..18.

Travis Sherman, Summation of Glaisher- and Apéry-like Series, University of Arizona, May 23 2000, p. 14, (3.79)-(3.83). (Scanned local copy without blank on p.12, with permission from the author.)

Formula

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.)

f2(n) corresponds to the y values, so f2(n) = (1/9)*((2*n-1)!/((n-1)!)^2).

a(n) ~ Pi * 2^(2*n + 1/2) * n^n / (sqrt(3) * exp(n)). - Vaclav Kotesovec, Apr 13 2018

Example

Examples ((3.79) - (3.83)) at page 14 in Links section as follows, respectively.
For n=1, f1(1) = 4/9, so a(1) = 4.
For n=2, f1(2) = 22/9, so a(2) = 22.
For n=3, f1(3) = 109/9, so a(3) = 218.
For n=4, f1(4) = 508/9, so a(4) = 3048.
For n=5, f1(5) = 4571/18, so a(5) = 54852.

Mathematica

RecurrenceTable[{a[n] == 2*((2*n - 1)*a[n-1] - (n-2)!), a[1] == 4}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 13 2018 *)
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 *)
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 *)

Crossrefs

Cf. A302770.

Sequence in context: A197999 A197963 A260296 * A137158 A025135 A125801

Adjacent sequences: A302766 A302767 A302768 * A302770 A302771 A302772

Keyword

nonn

Author

Detlef Meya, Apr 13 2018

Extensions

More terms from Vaclav Kotesovec, Apr 13 2018

Status

approved