1,1

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).

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

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+ ...

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]

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

sign,tabl,frac

Bradley Klee, Jun 09 2016

approved