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
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