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A217298 Triangle read by columns: T(n,k) = number of AVL trees of height n with k (leaf-) nodes, k>=1, A029837(k)<=n<A072649(k).

%I A217298 #42 Aug 05 2021 12:26:14
%S A217298 1,1,2,1,4,6,4,1,16,32,44,60,70,56,128,28,448,8,864,1,1552,2720,4288,
%T A217298 6312,9004,11992,4096,14372,22528,15400,67584,14630,159744,11968,
%U A217298 334080,8104,644992,4376,1195008,1820,2158912,560,3811904,120,6617184,16,11307904
%N A217298 Triangle read by columns: T(n,k) = number of AVL trees of height n with k (leaf-) nodes, k>=1, A029837(k)<=n<A072649(k).
%D A217298 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 239, Eq 79, A_5.
%D A217298 D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).
%H A217298 Alois P. Heinz, <a href="/A217298/b217298.txt">Columns k = 1..1500, flattened</a>
%H A217298 Ralf Hinze, <a href="https://www.cs.ox.ac.uk/ralf.hinze/publications/Brother12.pdf">Functional Pearls: Purely functional 1-2 brother trees</a>, Journal of Functional Programming, 19(6):633-644, 2009, DOI: <a href="http://dx.doi.org/10.1017/S0956796809007333">10.1017/S0956796809007333</a>.
%H A217298 R. C. Richards, <a href="http://dx.doi.org/10.1016/0020-0190(83)90085-6">Shape distribution of height-balanced trees</a>, Info. Proc. Lett., 17 (1983), 17-20.
%H A217298 Wikipedia, <a href="https://en.wikipedia.org/wiki/AVL_tree">AVL tree</a>
%H A217298 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%e A217298 There are 2 AVL trees of height 2 with 3 (leaf-) nodes:
%e A217298        o       o
%e A217298       / \     / \
%e A217298      o   N   N   o
%e A217298     / \         / \
%e A217298    N   N       N   N
%e A217298 Triangle begins:
%e A217298   1
%e A217298   . 1
%e A217298   . . 2 1
%e A217298   . . . . 4 6 4  1
%e A217298   . . . . . . . 16 32 44 60 70  56  28   8    1
%e A217298   . . . . . . .  .  .  .  .  . 128 448 864 1552 2720 ...
%Y A217298 Triangle read by rows gives: A143897.
%Y A217298 Row sums give: A029758.
%Y A217298 Column sums give: A006265.
%Y A217298 First elements of rows give: A174677.
%Y A217298 First, last elements of columns give: A217299, A217300.
%Y A217298 Row lengths give: 1+A008466(n).
%Y A217298 Column heights give: A217710(k).
%Y A217298 Cf. A029837, A072649.
%K A217298 nonn,look,tabf
%O A217298 1,3
%A A217298 _Alois P. Heinz_, Mar 17 2013