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A274076 T(n, m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact differential time dependence.

%I A274076 #24 Jun 13 2016 03:53:17
%S A274076 -2,2,-2,-4,8,-20,2,-58,14,-70,-4,16,-344,112,-28,4,-556,1064,-152,
%T A274076 308,-308,-8,10256,-3368,4576,-6248,2288,-1144,2,-1622,33398,-98794,
%U A274076 34606,-4862,2002,-1430,-4,6688,-187216,140384,-1242904,59488,-25168,77792,-48620
%N A274076 T(n, m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact differential time dependence.
%C A274076 Triangle read by rows ( see examples ). The denominators are given in A274078.
%C A274076 The rational triangle A273506 / A273507 gives the coefficients for an exact solution of the plane pendulum's phase space trajectory. Differential time dependence for this solution also adheres to the simple form of a triangular summation: dt = dQ(-1+ sum k^n * (T(n, m)/A274078(n, m)) * cos(Q)^(2(n+m)) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3...n. Expanding powers of cosine ( Cf. A273496 ) it is relatively easy to integrate dt ( cf. A274130 ). One period of motion takes Q through the range [ 0 , -2 pi]. Integrating dt over this domain gives another (Cf. A273506) calculation of the series expansion for Elliptic K ( see examples and Mathematica function dtToEllK ). For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).
%H A274076 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 A274076 The triangle T(n, m) begins:
%e A274076 n/m  1    2     3     4
%e A274076 ------------------------------
%e A274076 1  | -2
%e A274076 2  |  2, -2
%e A274076 3  | -4,  8,  -20
%e A274076 4  |  2, -58,  14,  -70
%e A274076 ------------------------------
%e A274076 The rational triangle T(n, m) / A274078(n, m) begins:
%e A274076 n/m    1        2         3       4
%e A274076 ------------------------------------------
%e A274076 1  | -2/3
%e A274076 2  |  2/15,   -2/3
%e A274076 3  | -4/315,   8/27,   -20/27
%e A274076 4  |  2/2835, -58/945,  14/27,  -70/81
%e A274076 ------------------------------------------
%e A274076 dt2(Q) = dQ(-1 - (2/3) cos(Q)^4 k +  ((2/15) cos(Q)^6  - (2/3) cos(Q)^8) k^2 ) + ...
%e A274076 dt2(Q) = dQ(-1 - (1/4) k - (9/64) k^2 + cosine series ) + ...
%e A274076 (2/Pi) K(k) ~ I2 = (1/(2 Pi)) Int dt2(Q) =  1 + (1/4) k + (9/64) k^2+ ...
%t A274076 R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
%t A274076 Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
%t A274076 RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
%t A274076 Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
%t A274076    Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
%t A274076 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 A274076 dtCoefficients[n_] :=  With[{dtn = dt[n]}, Function[{a}, Coefficient[ Coefficient[dtn, k^a], Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
%t A274076 dtToEllK[NMax_] := ReplaceAll[-dt[NMax], {Cos[Q]^n_ :> Divide[Binomial[n, n/2], (2^(n))], k^n_ /; n > NMax -> 0} ]
%t A274076 Flatten[Numerator[dtCoefficients[10]]]
%t A274076 dtToEllK[5]
%Y A274076 Denominators: A274078. Phase Space Trajectory: A273506, A273507. Time Dependence: A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.
%K A274076 sign,tabl,frac
%O A274076 1,1
%A A274076 _Bradley Klee_, Jun 09 2016