A273507 T(n, m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact phase space trajectory.

6, 45, 72, 630, 30, 144, 14175, 56700, 3240, 10368, 467775, 42525, 45360, 3888, 62208, 42567525, 2910600, 145800, 272160, 31104, 746496, 1277025750, 3831077250, 471517200, 729000, 13996800, 559872, 497664, 97692469875, 114932317500, 10945935000, 20207880000, 4199040, 124416, 746496, 23887872

Offset: 1

Author

Bradley Klee, May 23 2016

Keywords

• nonn
• tabl
• frac

Comments

Triangle read by rows ( see example ). The numerator triangle is A274076.

Comments of A273506 give a definition of the fraction triangle, which determines an arbitrary-precision solution to the simple pendulum 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  | 6
2  | 45,    72
3  | 630,   30,    144
4  | 14175, 56700, 3240, 10368
------------------------------
		

Links

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

Crossrefs

Numerators: A273506. Time Dependence: A274076, A274078, 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]]]]
RCoefficients[n_] :=  With[{Rn = ReplaceAll[R[n], RRules[n]]}, Function[{a},
    Coefficient[Coefficient[Rn/2/Sqrt[k], k^a],
       Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
Flatten[Denominator@RCoefficients[10]]