A305997 - OEIS

%I #18 Jun 28 2018 19:58:34

%S 44,45616,1669568,9778855936,3618728790016,10227537305460736,

%T 439851024281337856,283497572919345676288,262217569855510830645248,

%U 1411010811095175238386712576,51605826449550157277271425024,14612860454957563743068313616384

%N Define K(n) = Integral_{t=-1..1} t^(2n)*(1-t^2)^(2n)/(1+it)^(3n+1)dt and write K(n) = d(n)*Pi - a(n)/c(n) where a(n), d(n), c(n) are positive integers; sequence gives a(n).

%H Bradley Klee, <a href="/A305997/b305997.txt">Table of n, a(n) for n = 1..1000</a>

%H Frits Beukers, <a href="https://dspace.library.uu.nl/handle/1874/26398">A rational approach to Pi</a>, Nieuw archief voor wiskunde 5/1 No. 4, December 2000, p. 378.

%F Define G(x) = Sum_{n>0} A305997(n)/A305998(n)*x^n, and G^(n)(x) = d^n/dx^n G(x). Period G(x) satisfies a nonhomogeneous differential equation: -1097712 + 4292640*x + 3901584*x^2 - 224352*x^3 = Sum_{m=0..9, n=0..5} M_{m,n} x^m G^(n)(x), with integer matrix M as in A123178.

%t HermiteReduceRational[num_, den_, m_] := If[m > 1, Module[{cl = CoefficientList[num, t], deg, u, v, sol, c},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]

%t Numerator[ 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 *)

%t Numerator@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 *)

%Y Cf. A123178, A305998.

%K nonn

%O 1,1

%A _Bradley Klee_, Jun 16 2018