A242353 Number T(n,k) of two-colored rooted trees of order n and structure k; triangle T(n,k), n>=1, 1<=k<=A000081(n), read by rows.

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

Offset

1,1

Comments

The underlying partitions of n-1 (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, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 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 Strong-Coupling Limit, arXiv:1106.4938, 2011 (Physical Review B 85, 045105, Jan 2012)

E. Kalinowski and M. Paech, Table of two-colored 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, 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)

Example

Let {u, d} be a set of two colors, corresponding each with the up-spin and down-spin 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 two-colored 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 two-colored 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 two-colored 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 two-colored 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: A336350 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