A274078 T(n,m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact differential time dependence.

3, 15, 3, 315, 27, 27, 2835, 945, 27, 81, 155925, 2025, 2025, 135, 27, 6081075, 779625, 30375, 405, 243, 243, 638512875, 212837625, 654885, 42525, 8505, 1215, 729, 10854718875, 638512875, 58046625, 4465125, 127575, 3645, 729, 729

Offset: 1

Author

Bradley Klee, Jun 09 2016

Keywords

• nonn
• tabl
• frac

Comments

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).

Examples

			n\m|    1    2    3    4
---+---------------------
1  |    3;
2  |   15,   3;
3  |  315,  27,  27;
4  | 2835, 945,  27,  81;
		

Links

• Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.

Crossrefs

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

Programs

Mathematica

R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
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]]]
dtCoefficients[n_] :=  With[{dtn = dt[n]}, Function[{a}, Coefficient[ Coefficient[dtn, k^a], Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
Flatten[Denominator[dtCoefficients[10]]]