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 A276738 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. 5
 -1, -1, 5, -1, 12, -32, -1, 14, 7, -126, 231, -1, 16, 16, -160, -160, 1280, -1792, -1, 18, 18, -198, 9, -396, 1716, -66, 2574, -12870, 14586, -1, 20, 20, -240, 20, -480, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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). 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  | -1 2  | -1   5 3  | -1   12   -32 4  | -1   14    7   -126   231 5  | -1   16    16  -160  -160   1280  -1792 -------------------------------------------- R[1,Q] = -2*v_3*Q^3 R[2,Q] = -2*v_4*Q^4 + 10*v_3^2*Q^6 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) 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}. 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[#[], ReplaceAll[#[], rules]] & /@ # &, rules, n]][    Flatten[R[#, q] -> Expand[-ReplaceAll[ Coefficient[H, b^(# + 2)], {R[#, q] -> 0}]] & /@ Range[n]]]] basis[n_] :=  Times[Times @@ (v /@ #), Q^Total[#], 2] & /@ (IntegerPartitions[n] /. x_Integer :> x + 2) TriangleRow[n_, rules_] := With[{term = Expand[rules[[n, 2]]]},   Coefficient[term, #] & /@ basis[n]] With[{rules = RRules}, TriangleRow[#, rules] & /@ Range] CROSSREFS Arbitrary Oscillator: A276814, A276815, A276816, A276817. Pendulum: A273506, A273507, A274076, A274078, A274130, A274131, A038534, A056982, A000984, A001790, A038533, A046161, A273496. Sequence in context: A116923 A327797 A062264 * A094049 A286254 A322664 Adjacent sequences:  A276735 A276736 A276737 * A276739 A276740 A276741 KEYWORD sign,tabf AUTHOR Bradley Klee, Sep 16 2016 STATUS approved

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Last modified May 8 11:51 EDT 2021. Contains 343666 sequences. (Running on oeis4.)