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%I #31 Sep 08 2022 08:46:21
%S 2,5,24,166,1488,16344,212352,3184560,54132480,1028476800,21597649920,
%T 496742319360,12418518067200,335299508812800,9723679528550400,
%U 301433978206771200,9947319973149081600,348156178137427968000,12881778235397406720000,502389344778125156352000
%N a(n) = (2*n-1)*a(n-1) - (n-2)!, with a(1) = 2, n > 1.
%H Chai Wah Wu, <a href="/A303108/b303108.txt">Table of n, a(n) for n = 1..404</a>
%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. 15, (3.94) - (3.98).
%F a(n) = f1(n)*2*(n-1)!, where f1(n) corresponds to the x values such that Sum_{k>=0} 2^k/(binomial(2*k,k)*(k+n))) = x*Pi - 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) = (1/2^(n+2))*((2*n-1)!/((n-1)!)^2).
%F a(n) = 3*(n-1)*a(n-1) - (2n-3)*(n-2)*a(n-2) for n > 2. - _Chai Wah Wu_, Apr 20 2018
%e Examples ((3.94) - (3.98)) at page 15 in Links section as follows, respectively.
%e For n=1, f1(1) = 1, so a(1) = 2.
%e For n=2, f1(2) = 5/2, so a(2) = 5.
%e For n=3, f1(3) = 6, so a(3) = 24.
%e For n=4, f1(4) = 83/6, so a(4) = 166.
%e For n=5, f1(5) = 31, so a(5) = 1488.
%t RecurrenceTable[{a[n] == (2*n-1)*a[n-1] - (n-2)!, a[1] == 2}, a, {n, 1, 15}] (* _Altug Alkan_, Apr 20 2018 *)
%t nmax = 15; Table[CoefficientList[Expand[FunctionExpand[Table[Sum[ 2^j/(Binomial[2*j, j]*(j + m)), {j, 0, Infinity}], {m, 1, nmax}]]], Pi][[n, 2]]*2*(n-1)!, {n, 1, nmax}] (* _Vaclav Kotesovec_, Apr 20 2018 *)
%o (PARI) a=vector(20); a[1]=2; for(n=2, #a, a[n]=(2*n-1)*a[n-1] - (n-2)!); a
%o (Python)
%o A303108_list = [2,5]
%o for n in range(3,501):
%o A303108_list.append(3*(n-1)*A303108_list[-1]-(2*n-3)*(n-2)*A303108_list[-2]) # _Chai Wah Wu_, Apr 20 2018
%o (Magma) I:=[2,5]; [n le 2 select I[n] else 3*(n-1)*Self(n-1)- (2*n-3)*(n-2)*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Apr 20 2018
%Y Cf. A303109.
%K nonn
%O 1,1
%A _Detlef Meya_, Apr 18 2018
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