A198760 Number of initial spin configurations in two-colored rooted trees with n nodes.

2, 8, 32, 136, 596, 2712, 12642, 60234, 291840, 1434184, 7130640, 35807114, 181339236, 925139186, 4750176056, 24528421712, 127294780994, 663591911824, 3473315219722, 18246162722278, 96169600405626, 508413199626078, 2695245063006696, 14324688031734740

Offset

2,1

Comments

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

References

G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011

Links

Alois P. Heinz, Table of n, a(n) for n = 2..500

E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit, arXiv:1106.4938, 2011 (Physical Review B, January 2012).

M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012).

Formula

a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.6465426162329497128927135162..., c = 0.29201514711473716704145008728... . - Vaclav Kotesovec, Sep 12 2014

Maple

g:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(t*g(i-1$2, 2)+j-1, j)*g(n-i*j, i-1, t), j=0..n/i)))
    end:
a:= n-> 2*(g(n-1$2, 2) -g(n-1$2, 1)):
seq(a(n), n=2..30);  # Alois P. Heinz, May 12 2014

Mathematica

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 *)

Crossrefs

Cf. A000081, A038055, A198761, A225823, A245870.

Sequence in context: A006139 A150832 A150833 * A150834 A150835 A150836

Adjacent sequences: A198757 A198758 A198759 * A198761 A198762 A198763

Keyword

nonn

Author

N. J. A. Sloane, Oct 29 2011

Extensions

Terms a(8) and a(9) added by Martin Paech, Apr 16 2012

Term a(10) added by Martin Paech, Jul 30 2013

a(11)-a(25) from Alois P. Heinz, May 12 2014

Status

approved