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

Author

Bradley Klee, Jul 20 2015

Keywords

• nonn
• easy

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

• 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).

Crossrefs

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

Programs

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 *)

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

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