cp's OEIS Frontend

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).

%I A260253 #31 Jul 20 2023 15:43:12
%S A260253 1,0,4,1,9,2,16,4,25,7,36,10,49,14,64,19,81,24,100,30,121,37,144,44,
%T A260253 169,52,196,61,225,70,256,80,289,91,324,102,361,114,400,127,441,140,
%U A260253 484,154,529,169,576,184,625,200,676,217,729,234,784,252,841,271
%N 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).
%C A260253 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.
%C A260253 Harter et al. give only one second-order term, while Sadovskií et al. give only two (see links).
%H A260253 W. G. Harter, H. W. Galbraith, and C. W. Patterson, <a href="http://dx.doi.org/10.1063/1.436519">Energy level cluster analysis for E(v2) vibration rotation spectrum of spherical top molecules</a>, J. Chem. Phys, 69, 4888 (1978).
%H A260253 D. A. Sadovskií and B. I. Zhilinskií, <a href="http://dx.doi.org/10.1080/00268978800100891"> Qualitative analysis of vibration-rotation Hamiltonians for spherical top molecules</a>, Molecular Physics 65, 1 (1988).
%H A260253 N. J. A. Sloane, <a href="http://www.maa.org/programs/maa-awards/writing-awards/error-correcting-codes-and-invariant-theory-new-applications-of-a-nineteenth-century-technique">Error-correcting codes and invariant theory: new applications of a nineteenth-century technique</a>, American Mathematical Monthly (1977): 82-107.
%H A260253 Richard P. Stanley, <a href="http://dx.doi.org/10.1090/S0273-0979-1979-14597-X">Invariants of finite groups and their applications to combinatorics</a>, Bulletin of the American Mathematical Society 1.3 (1979): 475-511.
%H A260253 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A260253 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1,0,1,0,-2,0,1).
%F A260253 G.f.: (1 + 2 * x^2 + x^3 + 2 * x^4 + x^6 + x^7)/((1 - x^2)^3 *(1 + x^2 + x^4)).
%F A260253 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
%t A260253 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]
%t A260253 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 *)
%o A260253 (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
%Y A260253 Cf. A007980 (bisection), A002264, A260220, A000290 (bisection).
%K A260253 nonn,easy
%O A260253 0,3
%A A260253 _Bradley Klee_, Jul 20 2015