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 A274130 Irregular triangle T(n,m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact time dependence. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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). 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  | 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 ----------------------------------------------------------------------------- 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+... 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]] tToEllK[NMax_]:= Expand[((t[NMax] /. Q -> -2 Pi)/2/Pi) /. k^n_ /; n > NMax -> 0] Flatten[Numerator[-tCoefficients[10]]] tToEllK[5] CROSSREFS 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 KEYWORD nonn,tabf AUTHOR Bradley Klee, Jun 10 2016 STATUS approved

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Last modified January 15 13:34 EST 2021. Contains 340187 sequences. (Running on oeis4.)