A198761
Number of hopping sequences in four-colored rooted trees with n nodes, starting and ending with the same "initial state" from all of the (two-colored) rooted trees in A198760. See comments.
Original entry on oeis.org
2, 20, 648, 45472, 5644880, 1099056000, 310007943616, 119777421416192
Offset: 2
- G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
- 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)
A136794
Number of unlabeled marked rooted trees with n nodes.
Original entry on oeis.org
2, 8, 52, 376, 2998, 25256, 222128, 2013680, 18691522, 176743160, 1696546848, 16488151400, 161919802562, 1604274741608, 16016845623764, 160977882238968, 1627406260427490, 16537733701226936
Offset: 1
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 293 (4.1.60).
A225823
Number of hopping sequences in four-colored rooted trees with n nodes, starting and ending with the same "initial state" from all of the (two-colored) rooted trees in A198760. Compared to A198761, only one node color of the initial states is mobile on the tree (Falicov-Kimball model).
Original entry on oeis.org
1, 4, 54, 1568, 80680, 6510624, 761286848, 121944722176, 25668462562560
Offset: 2
- G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
- 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)
Term a(10) added by
Martin Paech, Sep 02 2013, calculated on a HP Integrity Superdome 2-32s by courtesy of Hewlett-Packard Development Company, L.P.
A242354
Number T(n,k) of four-colored rooted trees of order n and structure k; triangle T(n,k), n>=1, 1<=k<=A000081(n), read by rows.
Original entry on oeis.org
4, 16, 64, 40, 256, 160, 256, 80, 1024, 640, 1024, 320, 1024, 640, 544, 640, 140, 4096, 2560, 4096, 1280, 4096, 2560, 2176, 2560, 560, 4096, 2560, 4096, 1280, 4096, 2560, 2560, 1600, 2176, 1280, 224, 16384, 10240, 16384, 5120, 16384, 10240, 8704, 10240, 2240
Offset: 1
Let {h, u, d, p} be a set of four colors, corresponding to the four possible "states" of each tree node (lattice site) in the underlying physical problem, namely its occupation with no electron (hole), with one up-spin electron, with one down-spin electron, or with one up-spin and one down-spin electron (pair). (We consider each rooted tree as a cutout of the Bethe lattice in infinite dimensions.) Then for
n = 1 with A000081(1) = 1
h(), u(), d(), p() are the 4 four-colored trees of the first and only structure k = 1 (sum is 4 = A136793(1)); for
n = 2 with A000081(2) = 1
h(h()), h(u()), h(d()), h(p()),
u(h()), u(u()), u(d()), u(p()),
d(h()), d(u()), d(d()), d(p()),
p(h()), p(u()), p(d()), p(p()) are the 16 four-colored trees of the first and only structure k = 1 (sum is 16 = A136793(2)); for
n = 3 with A000081(3) = 2
h(h(h())), h(h(u())), h(h(d())), h(h(p())),
h(u(h())), ...
..., p(d(p())),
p(p(h())), p(p(u())), p(p(d())), p(p(p())) are the 64 four-colored trees of the structure k = 1 and
h(h()h()), h(h()u()), h(h()d()), h(h()p()),
h(u()u()), h(u()d()), h(u()p()),
h(d()d()), h(d()p()),
h(p()p()),
...,
p(h()h()), p(h()u()), p(h()d()), p(h()p()),
p(u()u()), p(u()d()), p(u()p()),
p(d()d()), p(d()p()),
p(p()p()) are the 40 four-colored trees of the structure k = 2 (sum is 104 = A136793(3)).
Triangle T(n,k) begins:
4;
16;
64, 40;
256, 160, 256, 80;
1024, 640, 1024, 320, 1024, 640, 544, 640, 140;
- G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
- Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.
- Martin Paech, Rows n = 1..10, flattened
- 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 85, 045105, Jan 2012)
- E. Kalinowski and M. Paech, Table of four-colored Butcher trees B(n,k,m) up to order n = 4.
- 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)
Total number of elements up to and including row n is
A087803.
Showing 1-4 of 4 results.
Comments