A260253 Number of symmetry-allowed, linearly-independent terms at n-th order in the expansion of E x (e+a) rovibrational perturbation matrix H(Jx,Jy,Jz).

1, 0, 4, 1, 9, 2, 16, 4, 25, 7, 36, 10, 49, 14, 64, 19, 81, 24, 100, 30, 121, 37, 144, 44, 169, 52, 196, 61, 225, 70, 256, 80, 289, 91, 324, 102, 361, 114, 400, 127, 441, 140, 484, 154, 529, 169, 576, 184, 625, 200, 676, 217, 729, 234, 784, 252, 841, 271

Offset

0,3

Comments

a(n) are also coefficients in a Molien Series for G = H x T x D3, where H is Hermitian conjugacy, T is Time-reversal, and D3 is triangular Dihedral. |G| = 24.

Harter et al. give only one second-order term, while Sadovskií et al. give only two (see links).

Links

Table of n, a(n) for n=0..57.

W. G. Harter, H. W. Galbraith, and C. W. Patterson, Energy level cluster analysis for E(v2) vibration rotation spectrum of spherical top molecules, J. Chem. Phys, 69, 4888 (1978).

D. A. Sadovskií and B. I. Zhilinskií, Qualitative analysis of vibration-rotation Hamiltonians for spherical top molecules, Molecular Physics 65, 1 (1988).

N. J. A. Sloane, Error-correcting codes and invariant theory: new applications of a nineteenth-century technique, American Mathematical Monthly (1977): 82-107.

Richard P. Stanley, Invariants of finite groups and their applications to combinatorics, Bulletin of the American Mathematical Society 1.3 (1979): 475-511.

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,0,1,0,-2,0,1).

Formula

G.f.: (1 + 2 * x^2 + x^3 + 2 * x^4 + x^6 + x^7)/((1 - x^2)^3 *(1 + x^2 + x^4)).

a(n)= 2*a(n-2) -a(n-4) +a(n-6) -2*a(n-8) +a(n-10). - R. J. Mathar, Jul 20 2023

Mathematica

D[(1 + 2 x^2 + x^3 + 2 x^4 + x^6 + x^7)/((1 - x^2)^3*(1 + x^2 + x^4)), {x, #}]/#!/.x -> 0 & /@ Range[0, 30]
CoefficientList[Series[(1 + 2 x^2 + x^3 + 2 x^4 + x^6 + x^7)/((1 - x^2)^3 (1 + x^2 + x^4)), {x, 0, 70}], x] (* Vincenzo Librandi, Jul 22 2015 *)

Prog

(PARI) Vec((1 + 2 * x^2 + x^3 + 2 * x^4 + x^6 + x^7)/((1 - x^2)^3 *(1 + x^2 + x^4)) + O(x^90)) \\ Michel Marcus, Aug 05 2015

Crossrefs

Cf. A007980 (bisection), A002264, A260220, A000290 (bisection).

Sequence in context: A306744 A304526 A333352 * A376783 A261981 A153265

Adjacent sequences: A260250 A260251 A260252 * A260254 A260255 A260256

Keyword

nonn,easy

Author

Bradley Klee, Jul 20 2015

Status

approved