A274131 Irregular triangle T(n,m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact time dependence.

6, 48, 96, 960, 160, 1536, 5760, 30720, 725760, 1935360, 34560, 165888, 23224320, 1161216, 4644864, 92897280, 4644864, 5806080, 663552, 21233664, 464486400, 3715891200, 232243200, 619315200, 11354112, 81749606400, 185794560, 2123366400, 26542080, 70778880

Offset

1,1

Comments

Irregular triangle read by rows (see example). The row length sequence is 2*n = A005843(n), n >= 1.

The numerator triangle is A274130.

Comments of A274130 give a definition of the fraction triangle, which determines to arbitrary precision the 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).

Links

Table of n, a(n) for n=1..30.

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

Example

n\m  1      2      3          4       5       6
------------------------------------------------------
1  | 6     48
2  | 96    960    160      1536
3  | 5760  30720  725760   1935360  34560   165888
------------------------------------------------------
row 4: 23224320, 1161216, 4644864, 92897280, 4644864, 5806080, 663552, 21233664,
row 5: 464486400, 3715891200, 232243200, 619315200, 11354112, 81749606400, 185794560, 2123366400, 26542080, 70778880.

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]]]
t[n_] := Expand[ReplaceAll[Q TrigReduce[dt[n]], Cos[x_ Q] :> (1/x/Q) Sin[x Q]]]
tCoefficients[n_] := With[{tn = t[n]}, Function[{a}, Coefficient[Coefficient[tn, k^a], Sin[2 # Q] ] & /@ Range[2 a]] /@ Range[n]]
Flatten[Denominator[-tCoefficients[10]]]

Crossrefs

Numerators: A274130. Phase Space Trajectory: A273506, A273507. Time Dependence: A274076, A274078. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

Sequence in context: A323138 A000252 A078237 * A341683 A259121 A052651

Adjacent sequences: A274128 A274129 A274130 * A274132 A274133 A274134

Keyword

nonn,tabf

Author

Bradley Klee, Jun 10 2016

Status

approved