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 A073346 Table T(n,k) (listed antidiagonalwise in order T(0,0), T(1,0), T(0,1), T(2,0), T(1,1), ...) giving the number of rooted plane binary trees of size n and "contracted height" k. 12
 1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 1, 0, 8, 8, 0, 0, 0, 0, 0, 0, 12, 16, 0, 0, 0, 0, 0, 0, 2, 12, 40, 16, 0, 0, 0, 0, 0, 0, 2, 12, 80, 48, 0, 0, 0, 0, 0, 0, 0, 0, 12, 136, 144, 32, 0, 0, 0, 0, 0, 0, 0, 2, 20, 224, 384, 128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The height of binary trees is computed here in the same way as in A073345, except that whenever a complete binary tree of (2^k)-1 nodes with all its leaves at the same level, i.e., one of the following trees: ____________________\/\/\/\/_ _____________\/__\/__\/__\/__ ______________\__/____\_ /___ ____.____\/____\/______\/____ etc. is encountered as a terminating subtree, it is regarded just a variant of . (an empty tree, a single leaf) and contributes nothing to the height of the tree. LINKS H. Bottomley and A. Karttunen, Notes concerning diagonals of the square arrays A073345 and A073346. FORMULA (See the Maple code below. Note that here we use the same convolution recurrence as with A073345, but only the initial conditions for the first two rows (k=0 and k=1) are different. Is there a nicer formula?) EXAMPLE The top-left corner of this square array: 1 1 0 1 0 0 0 1 ... 0 0 2 0 2 2 0 0 ... 0 0 0 4 4 8 12 12 ... 0 0 0 0 8 16 40 80 ... MAPLE A073346 := n -> A073346bi(A025581(n), A002262(n)); A073346bi := proc(n, k) option remember; local i, j; if(0 = k) then RETURN(A036987(n)); fi; if(0 = n) then RETURN(0); fi; 2 * add(A073346bi(n-i-1, k-1) * add(A073346bi(i, j), j=0..(k-1)), i=0..floor((n-1)/2)) + 2 * add(A073346bi(n-i-1, k-1) * add(A073346bi(i, j), j=0..(k-2)), i=(floor((n-1)/2)+1)..(n-1)) - (`mod`(n, 2))*(A073346bi(floor((n-1)/2), k-1)^2) - (`if`((1=k), 1, 0))*A036987(n); end; A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))), 2) - (n+1); A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))), 2); CROSSREFS Variant: A073345. The first row: A036987. Column sums: A000108. Diagonals: T(n, n) = A000007(n), T(n+1, n) = A000079(n), T(n+2, n) = A058922(n), T(n+3, n) = A074092(n) - [see the attached notes.]. A073430 gives the upper triangular region of this array. Used to compute A073431. Entries on row k are all divisible by 2^k, thus dividing them out yields the array/triangle A074079/A074080. Sequence in context: A233441 A255365 A256505 * A114099 A028613 A318381 Adjacent sequences:  A073343 A073344 A073345 * A073347 A073348 A073349 KEYWORD nonn,tabl AUTHOR Antti Karttunen, Jul 31 2002 EXTENSIONS Sequence number in comments corrected STATUS approved

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Last modified October 17 19:24 EDT 2019. Contains 328127 sequences. (Running on oeis4.)