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A274131
Irregular triangle T(n,m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact time dependence.
10
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
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
KEYWORD
nonn,tabf
AUTHOR
Bradley Klee, Jun 10 2016
STATUS
approved