%I A073345
%S A073345 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,
%T A073345 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,
%U A073345 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
%N 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.
%H A073345 H. Bottomley & A. Karttunen, <A HREF="http://www.research.att.com/~njas/sequences/A073345.txt">Notes concerning diagonals of the square arrays A073345 and A073346.</A>
%F A073345 (See the Maple code below. Is there a nicer formula?)
%e A073345 The top-left corner of this square array:
%e A073345 1 0 0 0 0 0 0 0 0 ...
%e A073345 0 1 0 0 0 0 0 0 0 ...
%e A073345 0 0 2 1 0 0 0 0 0 ...
%e A073345 0 0 0 4 6 6 4 1 0 ...
%e A073345 0 0 0 0 8 20 40 68 94 ...
%e A073345 E.g. we have A000108(3) = 5 binary trees built from 3 non-leaf (i.e. branching) nodes:
%e A073345 _______________________________3
%e A073345 ___\/__\/____\/__\/____________2
%e A073345 __\/____\/__\/____\/____\/_\/__1
%e A073345 _\/____\/____\/____\/____\./___0
%e A073345 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.
%p A073345 A073345 := n -> A073345bi(A025581(n),A002262(n));
%p A073345 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;
%p A073345 A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);
%p A073345 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
%Y A073345 Variant: A073346. Column sums: A000108. Row sums: A001699.
%Y A073345 Diagonals: A073345(n,n) = A011782(n), A073345(n,n-1) = A014480(n-3), A073345(n+2,n) = A073773(n), A073345(n+3,n) = A073774(n) - Henry Bottomley (se16@btinternet.com) and AK, see the attached notes.
%Y A073345 A073429 gives the upper triangular region of this array. Cf. also A065329, A001263.
%K A073345 nonn,tabl,new
%O A073345 0,13
%A A073345 Antti Karttunen (my_firstname.my_surname@iki.fi) Jul 31 2002