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A143897
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Triangle T(n,k)=number of AVL trees of height n with k (leaf-) nodes, n>=0, fibonacci(n+2)<=k<=2^n.
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2
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1, 1, 2, 1, 4, 6, 4, 1, 16, 32, 44, 60, 70, 56, 28, 8, 1, 128, 448, 864, 1552, 2720, 4288, 6312, 9004, 11992, 14372, 15400, 14630, 11968, 8104, 4376, 1820, 560, 120, 16, 1, 4096, 22528, 67584, 159744, 334080, 644992, 1195008, 2158912, 3811904, 6617184
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 239, Eq 79, A_5.
R. C. Richards, Shape distribution of height-balanced trees, Info. Proc. Lett., 17 (1983), 17-20.
D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..1682
Index entries for sequences related to rooted trees
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FORMULA
| See program.
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EXAMPLE
| There are 2 AVL trees of height 2 with 3 (leaf-) nodes:
o o
/ \ / \
o N N o
/ \ / \
N N N N
Triangle begins:
1
. 1
. . 2 1
. . . . 4 6 4 1
. . . . . . . 16 32 44 60 70 56 28 8 1
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MAPLE
| T:= proc(n, k) local B, z; B:= proc (x, y, d) if d>=1 then x+B(x^2+2*x*y, x, d-1) else x fi end; if n=0 then if k=1 then 1 else 0 fi else coeff (B(z, 0, n), z, k) -coeff (B(z, 0, n-1), z, k) fi end: fib:= m-> (Matrix([[1, 1], [1, 0]])^m)[1, 2]: seq (seq (T(n, k), k=fib(n+2)..2^n), n=0..6);
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CROSSREFS
| Row sums give A029758. Column sums give A006265. Column sums of first 5 or 6 rows give A036662 or A134306. Cf. A000045, A000079.
Sequence in context: A064786 A193902 A043302 * A036662 A134306 A006265
Adjacent sequences: A143894 A143895 A143896 * A143898 A143899 A143900
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KEYWORD
| nonn,tabf
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AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 04 2008
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