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A274078 T(n,m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact differential time dependence. 10
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

LINKS

Table of n, a(n) for n=1..36.

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

EXAMPLE

n\m|    1    2    3    4

---+---------------------

1  |    3;

2  |   15,   3;

3  |  315,  27,  27;

4  | 2835, 945,  27,  81;

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]]]

CROSSREFS

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

Sequence in context: A331225 A331223 A009215 * A302782 A088558 A212203

Adjacent sequences:  A274075 A274076 A274077 * A274079 A274080 A274081

KEYWORD

nonn,tabl,frac

AUTHOR

Bradley Klee, Jun 09 2016

STATUS

approved

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Last modified January 29 08:09 EST 2020. Contains 331337 sequences. (Running on oeis4.)