login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A276815
Irregular triangle read by rows T(n,m), coefficients in power/Fourier series expansion of an arbitrary anharmonic oscillator's exact differential time dependence.
4
3, 4, -24, 5, -70, 210, 6, -96, -48, 960, -1920, 7, -126, -126, 1386, 1386, -12012, 18018, 8, -160, -160, 1920, -80, 3840, -17920, 640, -26880, 143360, -172032, 9, -198, -198, 2574, -198, 5148, -25740, 2574, 2574, -77220, 218790, -25740, 437580, -1662804, 1662804, 10, -240, -240, 3360, -240, 6720, -35840, -120, 6720, 3360
OFFSET
1,1
COMMENTS
The phase space trajectory A276738 has phase space angular velocity A276814, which allows expansion of dt = dx /(dx/dt) = dx(-1 + sum b^n*T(n,m)*f(n,m)); where the sum runs over n = 1, 2, 3 ... and m = 1, 2, 3, ... A000041(n). The basis functions f(n,m) are the same as in A276738. To obtain period K, we integrate the function of Q=cos[x] over a range of [2*pi,0]. All odd powers of Q integrate to zero, so the period is an expansion in E=(1/2)*b^2 (Cf. A276816). This sequence transforms into A274076/A274078 by setting v_i=0 for odd i, v_i=(-1)^(i/2-1)/2/(i!) otherwise, and (1/2)*b^2 = 2*k. 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 7
------------------------------------------------
1 | 3
2 | 4 -24
3 | 5 -70 210
4 | 6 -96 -48 960 -1920
5 | 7 -126 -126 1386 1386 -12012 18018
------------------------------------------------
MATHEMATICA
R[n_] := b Plus[1, Total[b^# R[#, q] & /@ Range[n]]]
Vp[n_] := Total[2 v[# + 2] q^(# + 2) & /@ Range[n]]
H[n_] := Expand[1/2*r^2 + Vp[n]]
RRules[n_] := With[{H = Series[ReplaceAll[H[n], {q -> R[n] Q, r -> R[n]}], {b, 0, n + 2}]}, Function[{rules},
Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
Flatten[R[#, q] -> Expand[-ReplaceAll[ Coefficient[H, b^(# + 2)], {R[#, q] -> 0}]] & /@ Range[n]]]]
xDot[n_] := Expand[Normal@Series[ReplaceAll[ Q^2 D[D[q[t], t]/q[t], t], {D[q[t], t] -> R[n] P, q[t] -> R[n] Q, r -> R[n], D[q[t], {t, 2}]
-> ReplaceAll[D[-(q^2/2 + Vp[n]), q], q -> R[n] Q]} ], {b, 0, n}] /. RRules[n] /. {P^2 -> 1 - Q^2}]
dt[n_] := Expand[Normal@Series[1/xDot[n], {b, 0, n}]]
basis[n_] := Times[Times @@ (v /@ #), Q^Total[#], 2] & /@ (IntegerPartitions[n] /. x_Integer :> x + 2)
TriangleRow[n_, fun_] := Coefficient[fun, b^n #] & /@ basis[n]
With[{dt10 = dt[10]}, TriangleRow[#, dt10] /. v[_] -> 0 & /@ Range[10]]
KEYWORD
sign,tabf
AUTHOR
Bradley Klee, Sep 18 2016
STATUS
approved