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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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..36.

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: A286325 A063947 A006086 * A131513 A204558 A123141

Adjacent sequences:  A273504 A273505 A273506 * A273508 A273509 A273510

KEYWORD

nonn,tabl,frac

AUTHOR

Bradley Klee, May 23 2016

STATUS

approved

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Last modified January 23 16:36 EST 2020. Contains 331172 sequences. (Running on oeis4.)