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 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). 5
 1, 15, 7, 495, 2145, 69615, 33915, 245157, 2523675, 150225075, 60480225, 187751655, 26397882693, 7073942421, 221214302595, 26306570659215, 362711807574025, 315526198564395, 154309366568181825, 495353332454332275, 13575552922770178725 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 1 16:44 EST 2021. Contains 349430 sequences. (Running on oeis4.)