cp's OEIS Frontend

A231482 The number of nonlinear normal modes for a fully resonant Hamiltonian system with n degrees of freedom.

%I A231482 #30 Oct 11 2023 11:30:05
%S A231482 1,6,39,284,2205,17730,145635,1213560,10218105,86717630,740526303,
%T A231482 6355522068,54771976597,473667151482,4108390253595,35725327438320,
%U A231482 311346430241265,2718678371881590,23780515097337495,208330621395422220,1827615453799100301
%N A231482 The number of nonlinear normal modes for a fully resonant Hamiltonian system with n degrees of freedom.
%C A231482 The n-th term is the number of complex solutions to the algebraic equation for periodic orbits for the Hamiltonian H_2 + H_4, where H_2 is the sum of (p_j^2+q_j^2) (j=1..n) and H_4 is a generic homogeneous quartic which is invariant under the Hamiltonian flow generated by H_2, so this is a Hamiltonian in normal form.
%H A231482 G. C. Greubel, <a href="/A231482/b231482.txt">Table of n, a(n) for n = 1..1000</a>
%H A231482 Khazhgali Kozhasov, Alan Muniz, Yang Qi, and Luca Sodomaco, <a href="https://arxiv.org/abs/2309.15105">On the minimal algebraic complexity of the rank-one approximation problem for general inner products</a>, arXiv:2309.15105 [math.AG], 2023. See p. 13.
%H A231482 D. van Straten, <a href="http://dx.doi.org/10.1088/0951-7715/2/3/005">A note on the number of periodic orbits near a resonant equilibrium point</a>, Nonlinearity 2 (1989) 445-458.
%F A231482 G.f. (for offset 0): (1-x)^(-3/2)*(1-9*x)^(-1/2).
%F A231482 Recurrence: (n-1)*a(n) = 2*(5*n-7)*a(n-1) - 9*(n-1)*a(n-2). - _Vaclav Kotesovec_, Feb 14 2014
%F A231482 a(n) ~ sqrt(2) * 3^(2*n+1) / (32*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 14 2014
%t A231482 CoefficientList[Series[(1-x)^(-3/2)*(1-9*x)^(-1/2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 14 2014 *)
%o A231482 (PARI) lista(nn) = {x = xx + xx*O(xx^nn); expr = (1-x)^(-3/2)*(1-9*x)^(-1/2); for (i=0, nn, print1(polcoeff(expr, i, xx), ", "););} \\ _Michel Marcus_, Nov 10 2013
%K A231482 nonn
%O A231482 1,2
%A A231482 _James Montaldi_, Nov 09 2013