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A273506
T(n,m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact phase space trajectory.
17
1, -1, 7, 1, -1, 11, -1, 319, -143, 715, 1, -26, 559, -221, 4199, -2, 139, -323, 6137, -2261, 52003, 1, -10897, 135983, -4199, 527459, -52003, 37145, -1, 15409, -317281, 21586489, -52877, 7429, -88711, 1964315, 1, -76, 269123, -100901, 274873, -8671, 227447, -227447, 39803225, -2, 466003, -213739, 522629, -59074189, 226061641, -10690009, 25701511, -42077695, 547010035
OFFSET
1,3
COMMENTS
Triangle read by rows ( see examples ). The phase space trajectory of a simple pendulum can be written as (q,p) = (R(Q)cos(Q),R(Q)sin(Q)), with scaled, canonical coordinates q and p. The present triangle and A273507 determine a power / Fourier series of R(Q): R(Q) = sqrt(4 *k) * (1 + sum k^n * (A273506(n,m)/A273507(n,m)) * cos(Q)^(2(n+m)) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3...n. The period of an oscillator can be computed by T(k) = dA/dE, where A is the phase area enclosed by the phase space trajectory of conserved, total energy E. As we choose expansion parameter "k" proportional to E, the series expansion of the complete elliptic integral of the first kind follows from T(k) with very little technical difficulty ( see examples and Mathematica function R2ToEllK ). For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).
For some remarks on this pendulum problem and an alternative way to compute a(n,m) / A273507(n,m) using Lagrange inversion see the two W. Lang links. - Wolfdieter Lang, Jun 11 2016
EXAMPLE
n/m 1 2 3 4
------------------------------
1 | 1
2 | -1, 7
3 | 1, -1, 11
4 | -1, 319, -143, 715
------------------------------
R2(Q) = sqrt(4 k) (1 + (1/6) cos(Q)^4 k + (-(1/45) cos(Q)^6 + (7/72) cos(Q)^8) k^2)
R2(Q)^2 = 4 k + (4/3) cos(Q)^4 k^2 + ( -(8/45) cos(Q)^6 + (8/9) cos(Q)^8)k^3 + ...
I2 = (1/(2 Pi)) Int dQ (1/2)R2(Q)^2 = 2 k + (1/4) k^2 + (3/32) k^3 + ...
(2/Pi) K(k) ~ (1/2)d/dk(I2) = 1 + (1/4) k + (9/64) k^2 + ...
From Wolfdieter Lang, Jun 11 2016 (Start):
The rational triangle r(n,m) = a(n, m) / A273507(n,m) begins:
n\m 1 2 3 4 ...
1: 1/6
2: -1/45 7/72
3: 1/630 -1/30 11/144
4: -1/14175 319/56700 -143/3240 715/10368
... ,
row n = 5: 1/467775 -26/42525 559/45360 -221/3888 4199/62208,
row 6: -2/42567525 139/2910600 -323/145800 6137/272160 -2261/31104 52003/746496,
row 7: 1/1277025750 -10897/3831077250 135983/471517200 -4199/729000 527459/13996800 -52003/559872 37145/497664,
row 8:
-1/97692469875 15409/114932317500 -317281/10945935000 21586489/20207880000 -52877/4199040 7429/124416 -88711/746496 1964315/23887872.
... (End)
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]]]]
RCoefficients[n_] := With[{Rn = ReplaceAll[R[n], RRules[n]]}, Function[{a},
Coefficient[Coefficient[Rn/2/Sqrt[k], k^a],
Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
R2ToEllK[NMax_] := D[Expand[(2)^(-2) ReplaceAll[R[NMax], RRules[NMax]]^2] /. {Cos[Q]^n_ :> Divide[Binomial[n, n/2], (2^(n))], k^n_ /; n > NMax -> 0}, k]
Flatten[Numerator@RCoefficients[10]]
R2ToEllK[10]
CROSSREFS
Denominators: A273507. Time Dependence: A274076, A274078, A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.
Sequence in context: A358450 A183352 A217510 * A364093 A287326 A131065
KEYWORD
sign,tabl,frac
AUTHOR
Bradley Klee, May 23 2016
STATUS
approved