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%I #33 Apr 12 2018 02:53:11
%S 0,0,1,46,5547,1241628,447041205,236032398090,171832342201695,
%T 164958902421881400,201909733543855989225,306902783112141390315750,
%U 567156347906609067457417875,1252281213909270325492296700500,3255931157464155279716218995508125
%N a(n) = 8*(n-1)*(2*n-3)*a(n-1) + ((-1)^n)*(n-1)*Product_{k=0..n-3} (2*k+1)^2 with a(0) = 0.
%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, (3.48) - (3.52).
%F a(n) = (-1)^n*(f2(n-1)/8)*Product_{k=0..n-2} (2*k+1)^2 where f2(n) corresponds to the y values such that Sum_{k>=0} (-1)^k/(binomial(2*k,k)*(2*k+(2*n+1))) = x*sqrt(5)*log((1+sqrt(5))/2) + y. (See examples for connection with a(n) in terms of material at Links section).
%e Examples ((3.48) - (3.52)) at page 11 in Links section as follows, respectively.
%e For n=0, f2(0) = 0, so a(1) = 0.
%e For n=1, f2(1) = 8, so a(2) = 1.
%e For n=2, f2(2) = -368/9, so a(3) = 46.
%e For n=3, f2(3) = 14792/75, so a(4) = 5547.
%e For n=4, f2(4) = -3311008/3675, so a(5) = 1241628.
%t RecurrenceTable[{a[m+1] == 8*m*(2*m - 1)*a[m] + (-1)^(m + 1)*m * Product[(2*k + 1)^2, {k, 0, m - 2}], a[0] == 0}, a, {m, 0, 15}] (* _Vaclav Kotesovec_, Apr 11 2018 *)
%t nmax = 15; Flatten[{0, 0, Table[CoefficientList[TrigToExp[Expand[FunctionExpand[ Table[FullSimplify[Sum[(-1)^j/(Binomial[2*j, j]*(2*j + (2*m + 1))), {j, 0, Infinity}]]*(-1)^(m + 1)/8 * Product[(2*k + 1)^2, {k, 0, m - 1}], {m, 1, nmax}]]]], Log[1/2 + Sqrt[5]/2]][[n, 1]], {n, 1, nmax}]}] (* _Vaclav Kotesovec_, Apr 11 2018 *)
%o (PARI) a=vector(20); a[1]=0; for(n=2, #a, a[n]=8*(n-1)*(2*n-3)*a[n-1]+(-1)^n*(n-1)*prod(k=0, n-3, (2*k+1)^2)); concat(0, a) \\ _Altug Alkan_, Apr 01 2018
%Y Cf. A302113.
%K nonn
%O 0,4
%A _Detlef Meya_, Apr 01 2018
%E More terms from _Altug Alkan_, Apr 01 2018
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