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%I #46 Jun 19 2014 10:33:18
%S 2,4,8,6,16,12,16,8,32,24,32,16,32,24,20,24,10,64,48,64,32,64,48,40,
%T 48,20,64,48,64,32,64,48,48,36,40,32,12,128,96,128,64,128,96,80,96,40,
%U 128,96,128,64,128,96,96,72,80,64,24,128,96,128,64,128,96,80
%N 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.
%C 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.,
%C n = 1
%C {0} |-> () |-> 10_2
%C n = 2
%C {1} |-> (()) |-> 1100_2
%C n = 3
%C {2} > {1, 1} |-> ((())) > (()()) |-> 111000_2 > 110100_2
%C n = 4
%C {3} > {1, 2} > {1, 1, 1} |-> (((()))) > ((()())) > (()(())) > (()()()) |-> 11110000_2 > 11101000_2 > 11011000_2 > 11010100_2
%C The decimal equivalents of the binary encoded rooted trees in row n are the descending ordered elements of row n in A216648.
%D G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
%D Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.
%H Martin Paech, <a href="/A242353/b242353.txt">Rows n = 1..14, flattened</a>
%H E. Kalinowski and W. Gluza, <a href="http://arxiv.org/abs/1106.4938">Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit</a>, arXiv:1106.4938, 2011 (Physical Review B 85, 045105, Jan 2012)
%H E. Kalinowski and M. Paech, <a href="/A242353/a242353.pdf">Table of two-colored Butcher trees B(n,k,m) up to order n = 5</a>.
%H M. Paech, <a href="https://oeis.org/w/images/c/c0/A242353_MUSICALGORITHMS_3047_240.mid">A sonification of this sequence</a>, created with MUSICALGORITHMS, using simple 'division operation' instead of modulo scaling (3047 elements, 240 bpm).
%H M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, <a href="http://www.dpg-verhandlungen.de/year/2012/conference/berlin/part/tt/session/45/contribution/91">Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit</a>, DPG Spring Meeting, Berlin, TT 45.91 (2012)
%e 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
%e n = 1 with A000081(1) = 1
%e u(), d() are the 2 two-colored trees of the first and only structure k = 1 (sum is 2 = A038055(1)); for
%e n = 2 with A000081(2) = 1
%e 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
%e n = 3 with A000081(3) = 2
%e 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
%e 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)).
%e Triangle T(n,k) begins:
%e 2;
%e 4;
%e 8, 6;
%e 16, 12, 16, 8;
%e 32, 24, 32, 16, 32, 24, 20, 24, 10;
%Y Row sums give A038055.
%Y Row length is A000081.
%Y Total number of elements up to and including row n is A087803.
%Y Cf. A216648.
%K nonn,tabf,hear
%O 1,1
%A _Martin Paech_, May 11 2014
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