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 A221857 Number A(n,k) of shapes of balanced k-ary trees with n nodes, where a tree is balanced if the total number of nodes in subtrees corresponding to the branches of any node differ by at most one; square array A(n,k), n>=0, k>=0, read by antidiagonals. 10
 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 1, 1, 0, 1, 1, 4, 3, 4, 1, 0, 1, 1, 5, 6, 1, 4, 1, 0, 1, 1, 6, 10, 4, 9, 4, 1, 0, 1, 1, 7, 15, 10, 1, 27, 1, 1, 0, 1, 1, 8, 21, 20, 5, 16, 27, 8, 1, 0, 1, 1, 9, 28, 35, 15, 1, 96, 81, 16, 1, 0, 1, 1, 10, 36, 56, 35, 6, 25, 256, 81, 32, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened Jeffrey Barnett, Counting Balanced Tree Shapes, 2007 Samuele Giraudo, Intervals of balanced binary trees in the Tamari lattice, arXiv:1107.3472 [math.CO], Apr 2012 EXAMPLE : A(2,2) = 2  : A(2,3) = 3      : A(3,3) = 3          : :   o     o   :   o    o    o   :   o      o      o   : :  / \   / \  :  /|\  /|\  /|\  :  /|\    /|\    /|\  : : o         o : o      o      o : o o    o   o    o o : :.............:.................:.....................: : A(3,4) = 6                                          : :    o        o        o        o       o        o    : :  /( )\    /( )\    /( )\    /( )\   /( )\    /( )\  : : o o      o   o    o     o    o o     o   o      o o : Square array A(n,k) begins:   1, 1, 1,  1,   1,   1,  1,  1,  1,   1,   1, ...   1, 1, 1,  1,   1,   1,  1,  1,  1,   1,   1, ...   0, 1, 2,  3,   4,   5,  6,  7,  8,   9,  10, ...   0, 1, 1,  3,   6,  10, 15, 21, 28,  36,  45, ...   0, 1, 4,  1,   4,  10, 20, 35, 56,  84, 120, ...   0, 1, 4,  9,   1,   5, 15, 35, 70, 126, 210, ...   0, 1, 4, 27,  16,   1,  6, 21, 56, 126, 252, ...   0, 1, 1, 27,  96,  25,  1,  7, 28,  84, 210, ...   0, 1, 8, 81, 256, 250, 36,  1,  8,  36, 120, ... MAPLE A:= proc(n, k) option remember; local m, r; if n<2 or k=1 then 1       elif k=0 then 0 else r:= iquo(n-1, k, 'm');       binomial(k, m)*A(r+1, k)^m*A(r, k)^(k-m) fi     end: seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n-1, k]; Binomial[k, m]*a[r+1, k]^m*a[r, k]^(k-m)]]]; Table[a[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 17 2013, translated from Maple *) CROSSREFS Columns k=1-10 give: A000012, A110316, A131889, A131890, A131891, A131892, A131893, A229393, A229394, A229395. Rows n=0+1, 2-3, give: A000012, A001477, A179865. Diagonal and upper diagonals give: A028310, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, A001288. Lower diagonals give: A000012, A000290, A092364(n) for n>1. Sequence in context: A201093 A131255 A198295 * A133607 A103631 A263191 Adjacent sequences:  A221854 A221855 A221856 * A221858 A221859 A221860 KEYWORD nonn,tabl,look AUTHOR Alois P. Heinz, Apr 10 2013 STATUS approved

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Last modified August 21 06:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)