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%I #15 Dec 29 2018 03:27:36
%S 6,48,96,960,160,1536,5760,30720,725760,1935360,34560,165888,23224320,
%T 1161216,4644864,92897280,4644864,5806080,663552,21233664,464486400,
%U 3715891200,232243200,619315200,11354112,81749606400,185794560,2123366400,26542080,70778880
%N Irregular triangle T(n,m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact time dependence.
%C Irregular triangle read by rows (see example). The row length sequence is 2*n = A005843(n), n >= 1.
%C The numerator triangle is A274130.
%C 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).
%H Bradley Klee, <a href="http://arxiv.org/abs/1605.09102">Plane Pendulum and Beyond by Phase Space Geometry</a>, arXiv:1605.09102 [physics.class-ph], 2016.
%e n\m 1 2 3 4 5 6
%e ------------------------------------------------------
%e 1 | 6 48
%e 2 | 96 960 160 1536
%e 3 | 5760 30720 725760 1935360 34560 165888
%e ------------------------------------------------------
%e row 4: 23224320, 1161216, 4644864, 92897280, 4644864, 5806080, 663552, 21233664,
%e row 5: 464486400, 3715891200, 232243200, 619315200, 11354112, 81749606400, 185794560, 2123366400, 26542080, 70778880.
%t R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
%t Vq[n_] := Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
%t RRules[n_] := With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
%t Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
%t Flatten[R[#, Q] -> Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
%t 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 t[n_] := Expand[ReplaceAll[Q TrigReduce[dt[n]], Cos[x_ Q] :> (1/x/Q) Sin[x Q]]]
%t tCoefficients[n_] := With[{tn = t[n]},Function[{a}, Coefficient[Coefficient[tn, k^a], Sin[2 # Q] ] & /@ Range[2 a]] /@ Range[n]]
%t Flatten[Denominator[-tCoefficients[10]]]
%Y Numerators: A274130. Phase Space Trajectory: A273506, A273507. Time Dependence: A274076, A274078. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.
%K nonn,tabf
%O 1,1
%A _Bradley Klee_, Jun 10 2016
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