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A073345 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 height k. 10
1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 6, 8, 0, 0, 0, 0, 0, 0, 0, 4, 20, 0, 0, 0, 0, 0, 0, 0, 0, 1, 40, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 68, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 94, 152, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 114, 376 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

REFERENCES

Luo Jian-Jin, Catalan numbers in the history of mathematics in China, in Combinatorics and Graph Theory, (Yap, Ku, Lloyd, Wang, Editors), World Scientific, River Edge, NJ, 1995.

LINKS

Table of n, a(n) for n=0..96.

H. Bottomley and A. Karttunen, Notes concerning diagonals of the square arrays A073345 and A073346.

Andrew Odlyzko, Analytic methods in asymptotic enumeration.

FORMULA

(See the Maple code below. Is there a nicer formula?)

This table was known to the Chinese mathematician Ming An-Tu, who gave the following recurrence in the 1730s. a(0, 0) = 1, a(n, k) = Sum[a(n-1, k-1-i)( 2*Sum[ a(j, i), {j, 0, n-2}]+a(n-1, i) ), {i, 0, k-1}]. - David Callan, Aug 17 2004

The generating function for row n, T_n(x):=Sum[T(n, k)x^k, k>=0], is given by T_n = a(n)-a(n-1) where a(n) is defined by the recurrence a(0)=0, a(1)=1, a(n) = 1 + x a(n-1)^2 for n>=2. - David Callan, Oct 08 2005

EXAMPLE

The top-left corner of this square array is

1 0 0 0 0 0 0 0 0 ...

0 1 0 0 0 0 0 0 0 ...

0 0 2 1 0 0 0 0 0 ...

0 0 0 4 6 6 4 1 0 ...

0 0 0 0 8 20 40 68 94 ...

E.g. we have A000108(3) = 5 binary trees built from 3 non-leaf (i.e. branching) nodes:

_______________________________3

___\/__\/____\/__\/____________2

__\/____\/__\/____\/____\/_\/__1

_\/____\/____\/____\/____\./___0

The first four have height 3 and the last one has height 2, thus T(3,3) = 4, T(3,2) = 1 and T(3,any other value of k) = 0.

MAPLE

A073345 := n -> A073345bi(A025581(n), A002262(n));

A073345bi := proc(n, k) option remember; local i, j; if(0 = n) then if(0 = k) then RETURN(1); else RETURN(0); fi; fi; if(0 = k) then RETURN(0); fi; 2 * add(A073345bi(n-i-1, k-1) * add(A073345bi(i, j), j=0..(k-1)), i=0..floor((n-1)/2)) + 2 * add(A073345bi(n-i-1, k-1) * add(A073345bi(i, j), j=0..(k-2)), i=(floor((n-1)/2)+1)..(n-1)) - (`mod`(n, 2))*(A073345bi(floor((n-1)/2), k-1)^2); 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);

MATHEMATICA

a[0, 0] = 1; a[n_, k_]/; k<n||k>2^n-1 := 0; a[n_, k_]/; 1 <= n <= k <= 2^n-1 := a[n, k] = Sum[a[n-1, k-1-i](2Sum[ a[j, i], {j, 0, n-2}]+a[n-1, i]), {i, 0, k-1}]; Table[a[n, k], {n, 0, 9}, {k, 0, 9}]

(* or *) a[0] = 0; a[1] = 1; a[n_]/; n>=2 := a[n] = Expand[1 + x a[n-1]^2]; gfT[n_] := a[n]-a[n-1]; Map[CoefficientList[ #, x, 8]&, Table[gfT[n], {n, 9}]/.{x^i_/; i>=9 ->0}] (Callan)

CROSSREFS

Variant: A073346. Column sums: A000108. Row sums: A001699.

Diagonals: A073345(n, n) = A011782(n), A073345(n+3, n+2) = A014480(n), A073345(n+2, n) = A073773(n), A073345(n+3, n) = A073774(n) - Henry Bottomley and AK, see the attached notes.

A073429 gives the upper triangular region of this array. Cf. also A065329, A001263.

Sequence in context: A221836 A086077 A178408 * A216511 A138088 A112765

Adjacent sequences:  A073342 A073343 A073344 * A073346 A073347 A073348

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Jul 31 2002

STATUS

approved

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Last modified August 1 00:45 EDT 2014. Contains 245104 sequences.