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%I #23 Jan 10 2018 20:27:02
%S -1,-1,5,-1,12,-32,-1,14,7,-126,231,-1,16,16,-160,-160,1280,-1792,-1,
%T 18,18,-198,9,-396,1716,-66,2574,-12870,14586,-1,20,20,-240,20,-480,
%U 2240,-240,-240,6720,-17920,2240,-35840,129024,-122880,-1,22,22,-286,22,-572,2860,11,-572,-286,8580,-24310,-286,4290,8580,-97240,184756,715
%N Irregular triangle read by rows: T(n,m) = coefficients in a power/Fourier series expansion of an arbitrary anharmonic oscillator's exact phase space trajectory.
%C Irregular triangle read by rows (see examples). Consider an arbitrary anharmonic oscillator with Hamiltonian energy: H=(1/2)*b^2=(1/2)*(p^2+q^2) + Sum_{i=3} 2*v_i*q^i, and a stable minimum at (p,q)=(0,0). The phase space trajectory can be written in polar phase space coordinates as (q,p) = (R(x)cos(x),R(x)sin(x))=(R(Q)Q,R(Q)P). The present triangle determines a power / Fourier series of R(Q): R(Q) = b * (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 constructed from partitions of "n" listed in reverse lexicographic order. Partition n=(z_1+z_2+...z_j) becomes 2*Q^((z_1+2)+(z_2+2)+...(z_j+2))*v_{z_1+2}*v_{z_2+2}*...*v_{z_j+2} (see examples). This sequence transforms into A273506/A273507 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).
%H Bradley Klee, <a href="http://arxiv.org/abs/1605.09102">Plane Pendulum and Beyond by Phase Space Geometry</a>, arXiv:1605.09102 [physics.class-ph], 2016.
%e n/m 1 2 3 4 5 6 7
%e --------------------------------------------
%e 1 | -1
%e 2 | -1 5
%e 3 | -1 12 -32
%e 4 | -1 14 7 -126 231
%e 5 | -1 16 16 -160 -160 1280 -1792
%e --------------------------------------------
%e R[1,Q] = -2*v_3*Q^3
%e R[2,Q] = -2*v_4*Q^4 + 10*v_3^2*Q^6
%e R[Q] = b*(1+b*(-2*v_3*Q^3)+b^2*(-2*v_4*Q^4 + 10*v_3^2*Q^6 ))+O(b^4)
%e Construct basis for R[4,Q]; List partitions: {{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}; Transform Plus 2: {{v_6}, {v_5, v_3}, {v_4, v_4}, {v_4, v_3, v_3}, {v_3, v_3, v_3, v_3}}; Multiply: {v_6, v_5*v_3, v_4^2, v_4*v_3^2, v_3^4}; don't forget power of Q and factor of 2: {2*v_6*Q^6, 2*v_5*v_3*Q^8, 2*v_4^2*Q^8, 2*v_4*v_3^2*Q^10, 2*v_3^4*Q^12}.
%t R[n_] := b Plus[1, Total[b^# R[#, q] & /@ Range[n]]]
%t Vp[n_] := Total[2 v[# + 2] q^(# + 2) & /@ Range[n]]
%t H[n_] := Expand[1/2*r^2 + Vp[n]]
%t RRules[n_] := With[{H = Series[ReplaceAll[H[n], {q -> R[n] Q, r -> R[n]}], {b, 0, n + 2}]}, Function[{rules},
%t Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
%t Flatten[R[#, q] -> Expand[-ReplaceAll[ Coefficient[H, b^(# + 2)], {R[#, q] -> 0}]] & /@ Range[n]]]]
%t basis[n_] := Times[Times @@ (v /@ #), Q^Total[#],2] & /@ (IntegerPartitions[n] /. x_Integer :> x + 2)
%t TriangleRow[n_, rules_] := With[{term = Expand[rules[[n, 2]]]},
%t Coefficient[term, #] & /@ basis[n]]
%t With[{rules = RRules[10]}, TriangleRow[#, rules] & /@ Range[10]]
%Y Arbitrary Oscillator: A276814, A276815, A276816, A276817.
%Y Pendulum: A273506, A273507, A274076, A274078, A274130, A274131, A038534, A056982, A000984, A001790, A038533, A046161, A273496.
%K sign,tabf
%O 1,3
%A _Bradley Klee_, Sep 16 2016
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