cp's OEIS Frontend

A260220 Number of symmetry-allowed, linearly-independent terms at n-th order in the expansion of T1 x t1 rovibrational perturbation matrix H(Jx,Jy,Jz).

%I A260220 #42 Aug 01 2015 10:58:55
%S A260220 1,1,3,2,6,4,10,6,15,9,21,12,28,16,36,20,45,25,55,30,66,36,78,42,91,
%T A260220 49,105,56,120,64,136,72,153,81,171,90,190,100,210,110,231,121,253,
%U A260220 132,276,144,300,156,325,169,351,182,378,196,406,210,435,225,465,240
%N A260220 Number of symmetry-allowed, linearly-independent terms at n-th order in the expansion of T1 x t1 rovibrational perturbation matrix H(Jx,Jy,Jz).
%C A260220 a(n) are also coefficients in a Molien Series for G = H x T x O, where H is Hermitian conjugacy, T is Time-reversal, and O is Octahedral. |G| = 96.
%C A260220 Harter et al. give some terms. Compare first four values of a(n) with Eq. 8 of Dhont, Sadovskií, Zhilinskií, and Boudon (see links).
%H A260220 Guillaume Dhont, D. A. Sadovskií, B. I. Zhilinskií, & Vincent Boudon, <a href="http://dx.doi.org/10.1006/jmsp.2000.8070">Analysis of the "unusual" vibrational components of triply degenerate vibrational mode v6 of Mo(CO)6 based on the classical interpretation of the effective rotation-vibration Hamiltonian</a>, J. Mol. Spectrosc. 201, 95-108 (2000) (<a href="http://purple.univ-littoral.fr/~dima/preprints/MoCO6paper.ps.gz">alternate copy</a>).
%H A260220 W. G. Harter, H. W. Galbraith, and C. W. Patterson, <a href="http://dx.doi.org/10.1063/1.436520">Centrifugal and Coriolis effects on level cluster patterns for T(v3) rovibrational bands in spherical top molecules</a>, J. Chem. Phys, 69, 4896 (1978).
%H A260220 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 A260220 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 A260220 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A260220 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0, 2, 0, 0, 0, -2, 0, 1).
%F A260220 G.f.: (1 + x + x^2)/((1 - x^2)^3*(1 + x^2)).
%F A260220 a( 4 n+1 ) = A000290(n+1);
%F A260220 a( 4 n+3 ) = 2*A000217(n+1);
%F A260220 a( 2 n ) = A000217(n+1).
%t A260220 (D[(1 + x + x^2)/((1 - x^2)^3 (1 + x^2)), {x, #}]/#! /. x -> 0) & /@
%t A260220 Range[0, 20]
%t A260220 CoefficientList[Series[(1 + x + x^2)/((1 - x^2)^3 (1 + x^2)), {x, 0, 70}], x] (* _Vincenzo Librandi_, Jul 22 2015 *)
%t A260220 LinearRecurrence[{0, 2, 0, 0, 0, -2, 0, 1}, {1, 1, 3, 2, 6, 4, 10, 6}, 60] (* _Robert G. Wilson v_, Jul 22 2015 *)
%o A260220 (PARI) Vec((1 + x + x^2)/((1 - x^2)^3*(1 + x^2)) + O(x^80)) \\ _Michel Marcus_, Jul 20 2015
%Y A260220 Cf. A001399, A260253.
%K A260220 nonn
%O A260220 0,3
%A A260220 _Bradley Klee_, Jul 19 2015
%E A260220 More terms from _Michel Marcus_, Jul 20 2015