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A273507
T(n, m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact phase space trajectory.
11
6, 45, 72, 630, 30, 144, 14175, 56700, 3240, 10368, 467775, 42525, 45360, 3888, 62208, 42567525, 2910600, 145800, 272160, 31104, 746496, 1277025750, 3831077250, 471517200, 729000, 13996800, 559872, 497664, 97692469875, 114932317500, 10945935000, 20207880000, 4199040, 124416, 746496, 23887872
OFFSET
1,1
COMMENTS
Triangle read by rows ( see example ). The numerator triangle is A274076.
Comments of A273506 give a definition of the fraction triangle, which determines an arbitrary-precision solution to the simple pendulum equations of motion. For more details see "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
------------------------------
1 | 6
2 | 45, 72
3 | 630, 30, 144
4 | 14175, 56700, 3240, 10368
------------------------------
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]]
Flatten[Denominator@RCoefficients[10]]
CROSSREFS
Numerators: A273506. Time Dependence: A274076, A274078, A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.
Sequence in context: A335387 A063947 A006086 * A131513 A204558 A354477
KEYWORD
nonn,tabl,frac
AUTHOR
Bradley Klee, May 23 2016
STATUS
approved