A302733 - OEIS

%I #24 Apr 18 2018 04:51:23

%S 0,4,104,6348,714576,128844180,34036296120,12392237356380,

%T 5948858298045600,3640849196296529700,2767092887264215797000,

%U 2556812689319248012075500,2822730283098431620059222000,3669554568237694851067285432500,5548370007316561558557204595815000

%N a(n) = 4*n*(2*n-1)*a(n-1) + (4/9)*n*Product_{k=0..n} (2*k-3)^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, (3.58) - (3.62).

%F a(n-1) = (f2(n)/(2*n-1)^2)*(Product_{k=0..n-1} (2*k+1)^2), where f2(n) corresponds to the y 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.)

%e Examples ((3.58) - (3.62)) at page 11 in Links section as follows, respectively.

%e For n=1, f2(1) = 0, so a(0) = 0.

%e For n=2, f2(2) = 4, so a(1) = 4.

%e For n=3, f2(3) = 104/9, so a(2) = 104.

%e For n=4, f2(4) = 2116/75, so a(3) = 6348.

%e For n=5, f2(5) = 238192/3675, so a(4) = 714576.

%t RecurrenceTable[{a[n] == 4*n*(2*n-1)*a[n-1] + (4/9)*n*Product[(2*k-3)^2, {k, 0, n}], a[0] == 0}, a, {n, 0, 20}] (* _Altug Alkan_, Apr 12 2018 *)

%t nmax = 15; 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)^2, {k, 0, m - 1}])), {m, 1, nmax}]]]], Pi][[n, 1]], {n, 1, nmax}] (* _Vaclav Kotesovec_, Apr 13 2018 *)

%o (PARI) a=vector(20); a[1]=4; for(n=2, #a, a[n] = 4*n*(2*n-1)*a[n-1]+(4/9)*n*prod(k=0, n, (2*k-3)^2)); concat(0, a) \\ _Altug Alkan_, Apr 12 2018

%Y Cf. A302117.

%K nonn

%O 0,2

%A _Detlef Meya_, Apr 12 2018

%E More terms from _Altug Alkan_, Apr 12 2018