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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 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 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 July 28 17:47 EDT 2021. Contains 346335 sequences. (Running on oeis4.)