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

1, 6, 39, 284, 2205, 17730, 145635, 1213560, 10218105, 86717630, 740526303, 6355522068, 54771976597, 473667151482, 4108390253595, 35725327438320, 311346430241265, 2718678371881590, 23780515097337495, 208330621395422220, 1827615453799100301

Offset: 1

Author

James Montaldi, Nov 09 2013

Keywords

• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Khazhgali Kozhasov, Alan Muniz, Yang Qi, and Luca Sodomaco, On the minimal algebraic complexity of the rank-one approximation problem for general inner products, arXiv:2309.15105 [math.AG], 2023. See p. 13.
• D. van Straten, A note on the number of periodic orbits near a resonant equilibrium point, Nonlinearity 2 (1989) 445-458.

Programs

Mathematica

CoefficientList[Series[(1-x)^(-3/2)*(1-9*x)^(-1/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 14 2014 *)

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

Formula

G.f. (for offset 0): (1-x)^(-3/2)*(1-9*x)^(-1/2).

Recurrence: (n-1)*a(n) = 2*(5*n-7)*a(n-1) - 9*(n-1)*a(n-2). - Vaclav Kotesovec, Feb 14 2014

a(n) ~ sqrt(2) * 3^(2*n+1) / (32*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2014