%I #12 Dec 29 2018 03:34:33

%S 3,15,3,315,27,27,2835,945,27,81,155925,2025,2025,135,27,6081075,

%T 779625,30375,405,243,243,638512875,212837625,654885,42525,8505,1215,

%U 729,10854718875,638512875,58046625,4465125,127575,3645,729,729

%N T(n,m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact differential time dependence.

%C Triangle read by rows (see example). Comments of A274076 give a definition of the fraction triangle, which determines to arbitrary precision the differential time dependence for the time-independent solution (cf. A273506, A273507) of the plane pendulum's equations of motion. For more details see "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

%H Bradley Klee, <a href="http://arxiv.org/abs/1605.09102">Plane Pendulum and Beyond by Phase Space Geometry</a>, arXiv:1605.09102 [physics.class-ph], 2016.

%e n\m| 1 2 3 4

%e ---+---------------------

%e 1 | 3;

%e 2 | 15, 3;

%e 3 | 315, 27, 27;

%e 4 | 2835, 945, 27, 81;

%t R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]

%t Vq[n_] := Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]

%t RRules[n_] := With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},

%t Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][

%t Flatten[R[#, Q] -> Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]

%t dt[n_] := With[{rules = RRules[n]}, Expand[Subtract[ Times[Expand[D[R[n] /. rules, Q]], Normal@Series[1/R[n], {k, 0, n}] /. rules, Cot[Q] ], 1]]]

%t dtCoefficients[n_] := With[{dtn = dt[n]}, Function[{a}, Coefficient[ Coefficient[dtn, k^a], Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]

%t Flatten[Denominator[dtCoefficients[10]]]

%Y Numerators: A274076. Phase Space Trajectory: A273506, A273507. Time Dependence: A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

%K nonn,tabl,frac

%O 1,1

%A _Bradley Klee_, Jun 09 2016