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%I #47 Sep 08 2022 08:46:21
%S 0,1,3,23,273,4353,86955,2085975,58396905,1868565825,67266342675,
%T 2690619247575,118386592164225,5682542674572225,295491902843612475,
%U 16547538653388717975,992852105745276401625,63542528577414336074625,4320891751365390890563875
%N a(n) = 4*(n-1)*a(n-1) - (1/3)*Product_{k=0..n-1} (2*k-3), with a(0) = 0.
%H Robert Israel, <a href="/A302117/b302117.txt">Table of n, a(n) for n = 0..366</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. 11, (3.58) - (3.62).
%F a(n) = (f1(n)/(2*n-1))*(2*Product_{k=0..n-1} (2*k+1)), where f1(n) corresponds to the x values such that Sum_{k>=0} 2^k/(binomial(2*k,k)*(2*k+(2*n-1))) = x*Pi + y. (See examples for connection with a(n) in terms of material at Links section).
%F From _Robert Israel_, Apr 12 2018: (Start)
%F 3*a(n+2) - (18*n+9)*a(n+1) + 12*n*(2*n-1)*a(n) = 0.
%F E.g.f.: (1-sqrt(1-2*x))/2 + (arctanh(sqrt(2-4*x)) - arctanh(sqrt(2)))*sqrt(2)/4. (End)
%e Examples ((3.58) - (3.62)) at page 11 in Links section as follows, respectively.
%e For n=1, f1(1) = 1/2, so a(1) = 1.
%e For n=2, f1(2) = 3/2, so a(2) = 3.
%e For n=3, f1(3) = 23/6, so a(3) = 23.
%e For n=4, f1(4) = 91/10, so a(4) = 273.
%e For n=5, f1(5) = 1451/70, so a(5) = 4353.
%p P:= 1: A[0]:= 0:
%p for n from 1 to 20 do
%p P:= P*(2*(n-1)-3);
%p A[n]:= 4*(n-1)*A[n-1]-P/3;
%p od:
%p seq(A[i],i=0..20); # _Robert Israel_, Apr 12 2018
%t RecurrenceTable[{a[n] == 4*(n-1)*a[n-1] - Product[2*k-3, {k, 0, n-1}]/3, a[0] == 0}, a, {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 02 2018 *)
%t nmax = 16; Flatten[{0, Table[CoefficientList[Expand[FunctionExpand[Simplify[ Table[Sum[2^j/(Binomial[2*j, j]*(2*j+(2*m-1))), {j, 0, Infinity}]/((2*m-1)/(2*Product[(2*k+1), {k, 0, m-1}])), {m, 1, nmax}]]]], Pi][[n, 2]], {n, 1, nmax}]}] (* _Vaclav Kotesovec_, Apr 02 2018 *)
%o (PARI) a=vector(20); a[1]=1; for(n=2, #a, a[n]=4*(n-1)*a[n-1]-(1/3)*prod(k=0, n-1, (2*k-3))); concat(0, a) \\ _Altug Alkan_, Apr 01 2018
%o (Magma) [n le 1 select n else 4*(n-1)*Self(n) - (1/3)*&*[2*k-3: k in [0..n-1]]: n in [0..20]]; // _Bruno Berselli_, Apr 12 2018
%K nonn
%O 0,3
%A _Detlef Meya_, Apr 01 2018
%E a(10)-a(11) corrected by and more terms from _Altug Alkan_, Apr 01 2018
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