A305998 Let K(n) = integral(t=-1,1, t^(2n)*(1-t^2)^(2n)/(1+it)^(3n+1) dt) and write K(n) = d(n)*Pi - b(n)/a(n) where a(n), b(n), d(n) are positive integers; sequence gives a(n).

1, 15, 7, 495, 2145, 69615, 33915, 245157, 2523675, 150225075, 60480225, 187751655, 26397882693, 7073942421, 221214302595, 26306570659215, 362711807574025, 315526198564395, 154309366568181825, 495353332454332275, 13575552922770178725

Offset

1,2

Links

Bradley Klee, Table of n, a(n) for n = 1..1000

Frits Beukers, A rational approach to Pi, Nieuw archief voor wiskunde 5/1 No. 4, December 2000, p. 378.

Mathematica

HermiteReduceRational[num_, den_, m_] := If[m > 1, Module[{cl = CoefficientList[num, t], deg, u, v, sol}, If[Length[cl] == 1, cl = PadRight[cl, 3]]; deg = Length[cl] - 1; u = Total[c[#]*t^(2 #) & /@ Range[0, deg/2 - 1]]; v = Plus[Total[-c[#]*(m - 1)/(2*# + 1) t^(2*# + 1) & /@ Range[0, deg/2 - 1]], c[-1] t]; sol = Solve@ MapThread[Equal, {cl, CoefficientList[Expand[Dot[{1 + t^2, 2 t}, {u, v}]], t]}]; Plus[ ReplaceAll[v/(m - 1)/den^(m - 1), sol[[1]]] /. t -> 1, HermiteReduceRational[ Expand@ReplaceAll[u+1/(m-1)*D[v, t], sol[[1]]], den, m - 1]]], 0]
Denominator[ HermiteReduceRational[ t^(2*#)*(1-t^2)^(2*#)*((1+I*t)^(3*#+1)+(1-I*t)^(3*#+1)), (1+t^2), 3*#+1]]&/@Range[20] (* Bradley Klee, Jun 18 2018 *)
Denominator@RecurrenceTable[ {64*(1+n)*(2+n)*(1+2*n)*(3+2*n)*(5+2*n)*(816+755*n+165*n^2)*a[n] -48*(2+n)*(3+2*n)*(5+2*n)*(4+3*n)*(2039+4103*n+2595*n^2+495*n^3)*a[n+1] +6*(5+2*n)*(4+3*n)*(5+3*n)*(893628+2406908*n+2163923*n^2+803750*n^3+106095*n^4)*a[n+2] -9*(3+n)*(4+3*n)*(5+3*n)*(7+3*n)*(8+3*n)*(226+425*n+165*n^2)*a[n+3]==0, a[0]==0, a[1]==44, a[2]==45616/15}, a, {n, 1, 5000}] (* Bradley Klee, Jun 25 2018 *)

Crossrefs

Cf. A123178, A305997.

Sequence in context: A131876 A325136 A126070 * A103241 A194707 A094501

Adjacent sequences: A305995 A305996 A305997 * A305999 A306000 A306001

Keyword

nonn,frac

Author

Bradley Klee, Jun 16 2018

Status

approved