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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. 10
-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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..45.

Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.

EXAMPLE

The triangle T(n, m) begins:

n/m  1    2     3     4

------------------------------

1  | -2

2  |  2, -2

3  | -4,  8,  -20

4  |  2, -58,  14,  -70

------------------------------

The rational triangle T(n, m) / A274078(n, m) begins:

n/m    1        2         3       4

------------------------------------------

1  | -2/3

2  |  2/15,   -2/3

3  | -4/315,   8/27,   -20/27

4  |  2/2835, -58/945,  14/27,  -70/81

------------------------------------------

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+ ...

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]]]

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]

CROSSREFS

Denominators: A274078. Phase Space Trajectory: A273506, A273507. Time Dependence: A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

Sequence in context: A102831 A262568 A183388 * A160179 A021822 A153986

Adjacent sequences:  A274073 A274074 A274075 * A274077 A274078 A274079

KEYWORD

sign,tabl,frac

AUTHOR

Bradley Klee, Jun 09 2016

STATUS

approved

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Last modified January 19 06:34 EST 2020. Contains 331033 sequences. (Running on oeis4.)