

A242353


Number T(n,k) of twocolored rooted trees of order n and structure k; triangle T(n,k), n>=1, 1<=k<=A000081(n), read by rows.


2



2, 4, 8, 6, 16, 12, 16, 8, 32, 24, 32, 16, 32, 24, 20, 24, 10, 64, 48, 64, 32, 64, 48, 40, 48, 20, 64, 48, 64, 32, 64, 48, 48, 36, 40, 32, 12, 128, 96, 128, 64, 128, 96, 80, 96, 40, 128, 96, 128, 64, 128, 96, 96, 72, 80, 64, 24, 128, 96, 128, 64, 128, 96, 80
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OFFSET

1,1


COMMENTS

The underlying partitions of n1 (cf. A000041) for the construction of the trees with n nodes are generated in descending order, the elements within a partition are sorted in ascending order, e.g.,
n = 1
{0} > () > 10_2
n = 2
{1} > (()) > 1100_2
n = 3
{2} > {1, 1} > ((())) > (()()) > 111000_2 > 110100_2
n = 4
{3} > {1, 2} > {1, 1, 1} > (((()))) > ((()())) > (()(())) > (()()()) > 11110000_2 > 11101000_2 > 11011000_2 > 11010100_2
The decimal equivalents of the binary encoded rooted trees in row n are the descending ordered elements of row n in A216648.


REFERENCES

G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
Eva Kalinowski, MottHubbardIsolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der PhilippsUniversität, 2002.


LINKS

Martin Paech, Rows n = 1..14, flattened
E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the StrongCoupling Limit, arXiv:1106.4938, 2011 (Physical Review B 85, 045105, Jan 2012)
E. Kalinowski and M. Paech, Table of twocolored Butcher trees B(n,k,m) up to order n = 5.
M. Paech, A sonification of this sequence, created with MUSICALGORITHMS, using simple 'division operation' instead of modulo scaling (3047 elements, 240 bpm).
M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Groundstate energy and beyond: Highaccuracy results for the Hubbard model on the Bethe lattice in the strongcoupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012)


EXAMPLE

Let {u, d} be a set of two colors, corresponding each with the upspin and downspin electrons in the underlying physical problem. (We consider each rooted tree as a cutout of the Bethe lattice in infinite dimensions.) Then for
n = 1 with A000081(1) = 1
u(), d() are the 2 twocolored trees of the first and only structure k = 1 (sum is 2 = A038055(1)); for
n = 2 with A000081(2) = 1
u(u()), u(d()), d(u()), d(d()) are the 4 twocolored trees of the first and only structure k = 1 (sum is 4 = A038055(2)); for
n = 3 with A000081(3) = 2
u(u(u())), u(u(d())), u(d(u())), u(d(d())), d(u(u())), d(u(d())), d(d(u())), d(d(d())) are the 8 twocolored trees of the structure k = 1 and
u(u()u()), u(u()d()), u(d()d()), d(u()u()), d(u()d()), d(d()d()) are the 6 twocolored trees of the structure k = 2 (sum is 14 = A038055(3)).
Triangle T(n,k) begins:
2;
4;
8, 6;
16, 12, 16, 8;
32, 24, 32, 16, 32, 24, 20, 24, 10;


CROSSREFS

Row sums give A038055.
Row length is A000081.
Total number of elements up to and including row n is A087803.
Cf. A216648.
Sequence in context: A253885 A140692 A048640 * A277331 A124510 A131886
Adjacent sequences: A242350 A242351 A242352 * A242354 A242355 A242356


KEYWORD

nonn,tabf,hear


AUTHOR

Martin Paech, May 11 2014


STATUS

approved



