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A274076
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T(n, m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact differential time dependence.
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10
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-2, 2, -2, -4, 8, -20, 2, -58, 14, -70, -4, 16, -344, 112, -28, 4, -556, 1064, -152, 308, -308, -8, 10256, -3368, 4576, -6248, 2288, -1144, 2, -1622, 33398, -98794, 34606, -4862, 2002, -1430, -4, 6688, -187216, 140384, -1242904, 59488, -25168, 77792, -48620
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OFFSET
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1,1
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COMMENTS
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Triangle read by rows ( see examples ). The denominators are given in A274078.
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).
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LINKS
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EXAMPLE
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The triangle T(n, m) begins:
n/m 1 2 3 4
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1 | -2
2 | 2, -2
3 | -4, 8, -20
4 | 2, -58, 14, -70
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The rational triangle T(n, m) / A274078(n, m) begins:
n/m 1 2 3 4
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1 | -2/3
2 | 2/15, -2/3
3 | -4/315, 8/27, -20/27
4 | 2/2835, -58/945, 14/27, -70/81
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dt2(Q) = dQ(-1 - (2/3) cos(Q)^4 k + ((2/15) cos(Q)^6 - (2/3) cos(Q)^8) k^2 ) + ...
dt2(Q) = dQ(-1 - (1/4) k - (9/64) k^2 + cosine series ) + ...
(2/Pi) K(k) ~ I2 = (1/(2 Pi)) Int dt2(Q) = 1 + (1/4) k + (9/64) k^2+ ...
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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]]]
dtCoefficients[n_] := With[{dtn = dt[n]}, Function[{a}, Coefficient[ Coefficient[dtn, k^a], Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
dtToEllK[NMax_] := ReplaceAll[-dt[NMax], {Cos[Q]^n_ :> Divide[Binomial[n, n/2], (2^(n))], k^n_ /; n > NMax -> 0} ]
Flatten[Numerator[dtCoefficients[10]]]
dtToEllK[5]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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