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%I #47 Apr 12 2018 04:50:01
%S 1,7,111,2661,85137,3405375,163457055,9153584685,585829284705,
%T 42179706471735,3374376483279375,296945129873855925,
%U 28506732454140858225,2964700174914415112175,332046419582508638982975,39845570349687578631280125,5100233004753819781450226625
%N a(n) = 8*(n-1)*a(n-1) + Product_{k=0..n-2} (2*k-1) with a(1) = 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. 10, (3.43) - (3.47).
%F a(n) = (f1(n)/(4*n-2))*9*Product_{k=1..n} (2*k-1) where f1(n) corresponds to the x values such that Sum_{k>=0} 1/(binomial(2*k,k)*(2*k+(2*n-1))) = x*Pi*sqrt(3) - y. (See examples for connection with a(n) in terms of material at Links section).
%e Examples ((3.43) - (3.47)) at page 10 in Links section as follows, respectively.
%e For n=1, f1(1) = 2/9, so a(1) = 1.
%e For n=2, f1(2) = 14/9, so a(2) = 7.
%e For n=3, f1(3) = 74/9, so a(3) = 111.
%e For n=4, f1(4) = 1774/45, so a(4) = 2661.
%e For n=5, f1(5) = 56758/315, so a(5) = 85137.
%t RecurrenceTable[{a[n + 1] == 8*n*a[n] + Product[(2*k - 1), {k, 0, n - 1}], a[1] == 1}, a, {n, 1, 20}] (* _Vaclav Kotesovec_, Mar 30 2018 *)
%t Table[FullSimplify[2^(-4 + 3 n) Sqrt[3] Gamma[n] + 2^(-2 - n) Gamma[-1 + 2 n] Hypergeometric2F1Regularized[1, -1/2 + n, 1 + n, 1/4]], {n, 1, 20}] (* _Vaclav Kotesovec_, Mar 30 2018 *)
%t nmax = 15; Table[1/Sqrt[3]*CoefficientList[Expand[FunctionExpand[Table[ FullSimplify[Sum[1/(Binomial[2*j, j]*(2*j + (2*m - 1))), {j, 0, Infinity}]] * 9 * Product[(2*k - 1), {k, 1, m}]/(4*m - 2), {m, 1, nmax}]]], Pi][[n, 2]], {n, 2, nmax}] (* _Vaclav Kotesovec_, Apr 12 2018 *)
%o (PARI) a=vector(20); a[1]=1; for(n=2, #a, a[n]=8*(n-1)*a[n-1] + prod(k=0, n-2, 2*k-1)); a \\ _Altug Alkan_, Mar 30 2018
%Y Cf. A000984, A301992.
%K nonn
%O 1,2
%A _Detlef Meya_, Mar 30 2018
%E More terms from _Vaclav Kotesovec_, Mar 30 2018
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