OFFSET
1,3
COMMENTS
Irregular triangle read by rows ( see examples ). The row length sequence is 2*n = A005843(n), n >= 1.The denominators are given in A274131.
The triangles A274076 and A274078 give the coefficients for the exact, differential time dependence of the plane pendulum's equations of motion. Integrating, we obtain time dependence as a Fourier sine series: t = -( (2/pi)K(k) Q + sum k^n * (T(n,m)/A274131(n,m)) * sin(2 m Q) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3,...,2 n. Combining the phase space trajectory and time dependence, it is possible to express Jacobian elliptic functions {cn,dn} in parametric form. For more details read "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 5 6 ...
-----------------------------------------
1 | 1 1
2 | 11 29 1 1
3 | 491 863 6571 4399 13 5
row n=4: 1568551, 28783, 45187, 312643, 4351, 1117, 17, 35,
row n=5: 25935757, 81123251, 2226193, 2440117, 16025, 34246631, 18161, 35443, 49, 7.
-----------------------------------------
The rational irregular triangle T(n, m) / A274131(n, m) begins:
n\m 1 2 3 4 5 6
-----------------------------------------------------------------------------
1 | 1/6, 1/48
2 | 11/96, 29/960, 1/160, 1/1536
3 | 491/5760, 863/30720, 6571/725760, 4399/1935360, 13/34560, 5/165888
row n=4: 1568551/23224320, 28783/1161216, 45187/4644864, 312643/92897280, 4351/4644864, 1117/5806080, 17/663552, 35/21233664,
row n=5: 25935757/464486400, 81123251/3715891200, 2226193/232243200, 2440117/619315200, 16025/11354112, 34246631/81749606400, 18161/185794560, 35443/2123366400, 49/26542080, 7/70778880.
-----------------------------------------------------------------------------
t1(Q) =-Q -(1/4)*k*Q -k*((1/6)*Sin[2*Q]+(1/48)*Sin[4*Q])+...
(2/Pi) K(k) ~ (1/(2 Pi)) t1(-2*Pi) = 1+(1/4)*k+...
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]]]]
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]]]
t[n_] := Expand[ReplaceAll[Q TrigReduce[dt[n]], Cos[x_ Q] :> (1/x/Q) Sin[x Q]]]
tCoefficients[n_] := With[{tn = t[n]}, Function[{a}, Coefficient[Coefficient[tn, k^a], Sin[2 # Q] ] & /@ Range[2 a]] /@ Range[n]]
tToEllK[NMax_]:= Expand[((t[NMax] /. Q -> -2 Pi)/2/Pi) /. k^n_ /; n > NMax -> 0]
Flatten[Numerator[-tCoefficients[10]]]
tToEllK[5]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Bradley Klee, Jun 10 2016
STATUS
approved