Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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