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A274130
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Irregular triangle T(n,m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact time dependence.
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10
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1, 1, 11, 29, 1, 1, 491, 863, 6571, 4399, 13, 5, 1568551, 28783, 45187, 312643, 4351, 1117, 17, 35, 25935757, 81123251, 2226193, 2440117, 16025, 34246631, 18161, 35443, 49, 7, 5301974777, 22870237, 1603483793, 23507881213, 122574691, 122330761339, 903325919, 1976751869, 956873, 18551, 35, 77
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OFFSET
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1,3
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COMMENTS
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Irregular triangle read by rows ( see examples ). The row length sequence is 2*n = A005843(n), n >= 1.The denominators are given in A274131.
The triangles A274076 and A274078 give the coefficients for the exact, differential time dependence of the plane pendulum's equations of motion. Integrating, we obtain time dependence as a Fourier sine series: t = -( (2/pi)K(k) Q + sum k^n * (T(n,m)/A274131(n,m)) * sin(2 m Q) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3,...,2 n. Combining the phase space trajectory and time dependence, it is possible to express Jacobian elliptic functions {cn,dn} in parametric form. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).
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LINKS
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Table of n, a(n) for n=1..42.
Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.
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EXAMPLE
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n\m 1 2 3 4 5 6 ...
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1 | 1 1
2 | 11 29 1 1
3 | 491 863 6571 4399 13 5
row n=4: 1568551, 28783, 45187, 312643, 4351, 1117, 17, 35,
row n=5: 25935757, 81123251, 2226193, 2440117, 16025, 34246631, 18161, 35443, 49, 7.
-----------------------------------------
The rational irregular triangle T(n, m) / A274131(n, m) begins:
n\m 1 2 3 4 5 6
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1 | 1/6, 1/48
2 | 11/96, 29/960, 1/160, 1/1536
3 | 491/5760, 863/30720, 6571/725760, 4399/1935360, 13/34560, 5/165888
row n=4: 1568551/23224320, 28783/1161216, 45187/4644864, 312643/92897280, 4351/4644864, 1117/5806080, 17/663552, 35/21233664,
row n=5: 25935757/464486400, 81123251/3715891200, 2226193/232243200, 2440117/619315200, 16025/11354112, 34246631/81749606400, 18161/185794560, 35443/2123366400, 49/26542080, 7/70778880.
-----------------------------------------------------------------------------
t1(Q) =-Q -(1/4)*k*Q -k*((1/6)*Sin[2*Q]+(1/48)*Sin[4*Q])+...
(2/Pi) K(k) ~ (1/(2 Pi)) t1(-2*Pi) = 1+(1/4)*k+...
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MATHEMATICA
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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]]
tToEllK[NMax_]:= Expand[((t[NMax] /. Q -> -2 Pi)/2/Pi) /. k^n_ /; n > NMax -> 0]
Flatten[Numerator[-tCoefficients[10]]]
tToEllK[5]
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CROSSREFS
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Denominators: A274131. Phase Space Trajectory: A273506, A273507. Time Dependence: A274076, A274078. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.
Sequence in context: A018944 A061086 A201633 * A034276 A245169 A323822
Adjacent sequences: A274127 A274128 A274129 * A274131 A274132 A274133
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KEYWORD
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nonn,tabf
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AUTHOR
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Bradley Klee, Jun 10 2016
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STATUS
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approved
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