|
%I #55 Nov 25 2014 09:10:27
%S 2,8,32,136,596,2712,12642,60234,291840,1434184,7130640,35807114,
%T 181339236,925139186,4750176056,24528421712,127294780994,663591911824,
%U 3473315219722,18246162722278,96169600405626,508413199626078,2695245063006696,14324688031734740
%N Number of initial spin configurations in two-colored rooted trees with n nodes.
%C Also the number of two-colored rooted trees that have for a given color of the root at least one nearest neighbor node of the root in the other color. - _Martin Paech_, Apr 16 2012
%D G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
%H Alois P. Heinz, <a href="/A198760/b198760.txt">Table of n, a(n) for n = 2..500</a>
%H E. Kalinowski and W. Gluza, <a href="http://arxiv.org/abs/1106.4938">Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit</a>, arXiv:1106.4938, 2011 (Physical Review B, January 2012).
%H M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, <a href="http://www.dpg-verhandlungen.de/year/2012/conference/berlin/part/tt/session/45/contribution/91">Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit</a>, DPG Spring Meeting, Berlin, TT 45.91 (2012).
%F a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.6465426162329497128927135162..., c = 0.29201514711473716704145008728... . - _Vaclav Kotesovec_, Sep 12 2014
%p g:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p binomial(t*g(i-1$2, 2)+j-1, j)*g(n-i*j, i-1, t), j=0..n/i)))
%p end:
%p a:= n-> 2*(g(n-1$2, 2) -g(n-1$2, 1)):
%p seq(a(n), n=2..30); # _Alois P. Heinz_, May 12 2014
%t g[n_, i_, t_] := g[n, i, t] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[t*g[i-1, i-1, 2]+j-1, j]*g[n-i*j, i-1, t], {j, 0, n/i}]]]; a[n_] := 2*(g[n-1, n-1, 2] - g[n-1, n-1, 1]) // FullSimplify; Table[a[n], {n, 2, 30}] (* _Jean-François Alcover_, Nov 25 2014, after _Alois P. Heinz_ *)
%Y Cf. A000081, A038055, A198761, A225823, A245870.
%K nonn
%O 2,1
%A _N. J. A. Sloane_, Oct 29 2011
%E Terms a(8) and a(9) added by _Martin Paech_, Apr 16 2012
%E Term a(10) added by _Martin Paech_, Jul 30 2013
%E a(11)-a(25) from _Alois P. Heinz_, May 12 2014
|
