Cheers, 5 new sequences: A073345 - A073346, A073429 - A073431,
and two edited lines from existing sequences:

First the edited lines, of A001699 and A065329,
only one %C line from the first, and one %Y-line from the second
are changed:

%I A001699 M3087 N1251
%S A001699 1,1,3,21,651,457653,210065930571,44127887745696109598901,
%C A001699 Approaches 1.5028368...^(2^n). Row sum of A065329 as square array. - Henry Bottomley (se16@btinternet.com), Oct 29 2001. Also row sum of square array A073345 (AK).


%I A065329
%S A065329 1,0,2,0,1,8,0,0,8,80,0,0,4,144,4160,0,0,1,168,13888,5632640,0,0,0,138,
%Y A065329 Row sums are A001699. Cf. A073345 (A073429).

-----------------------------------------------------------------------------------

Then the new sequences:

%I A073345
%S A073345 1,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,
%T A073345 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,
%U A073345 0,0,94,152,32,0,0,0,0,0,0,0,0,0,0,114,376,144,0,0,0,0,0,0,0
%N A073345
%N A073245 Table T(n,k) (listed antidiagonalwise in order T(0,0), T(1,0), T(0,1), T(2,0), T(1,1), ...) giving a number of rooted plane binary trees of size n and height k.
%F A073345 (See the Maple code below. Anybody for a nicer formula?)
%e A073345 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.
%R A073345
%O A073345 0,13
%K A073345 nonn,tabl,new
%A A073345 Antti Karttunen (my_firstname.my_surname@iki.fi) Jul 31 2002
%Y A073345 Variant: A073346. Column sums: A000108. Row sums: A001699. Super-diagonal T(n,n-1) = A014480 ?
%Y A073345 A073429 gives the upper triangular region of this array. Cf. also A065329, A001263.
%D A073345
%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);



%I A073346
%S A073346 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,
%T A073346 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,
%U A073346 2,20,224,384,128,0,0,0,0,0,0,0,0,0,16,328,912,448,64,0,0,0,0,0,0,0
%N A073246 Table T(n,k) (listed antidiagonalwise in order T(0,0), T(1,0), T(0,1), T(2,0), T(1,1), ...) giving a number of rooted plane binary trees of size n and "contracted height" k.
%C A073246 The height of binary trees is computed here otherwise as with A073245, but 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:
%C A073246 ____________________\/\/\/\/_
%C A073246 _____________\/__\/__\/__\/__
%C A073246 ______________\__/____\_ /___
%C A073246 ____.____\/____\/______\/____ etc.
%C A073246 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.
%F A073246 (See the Maple code below. Anybody for a nicer formula?)
%N A073346
%R A073346
%O A073346 0,8
%K A073346 nonn,tabl,new
%A A073346 Antti Karttunen (my_firstname.my_surname@iki.fi) Jul 31 2002
%Y A073346 Variant: A073345. The first row: A036987. Column sums: A000108. The diagonal over the super-diagonal T(n,n-2) = A058922 = 4*A001787 ?
%Y A073346 A073430 gives the upper triangular region of this array. Used to compute A073431.
%D A073346
%p A073346 A073346 := n -> A073346bi(A025581(n),A002262(n));
%p A073346 A036987 := n -> `if`(((2^floor_log_2(n+1)) = (n+1)),1,0);
%p A073346 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;
%p A073346 A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);
%p A073346 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);



%I A073429
%S A073429 1,0,1,0,0,2,0,0,1,4,0,0,0,6,8,0,0,0,6,20,16,0,0,0,4,40,56,32,0,0,0,1,68,152,144,64,0,0,0,0,
%T A073429 94,376,480,352,128,0,0,0,0,114,844,1440,1376,832,256,0,0,0,0,116,1744,4056,4736,3712,1920,
%U A073429 512,0,0,0,0,94,3340,10856,15248,14272,9600,4352,1024,0,0,0,0,60,5976,27672,47104,50784,40576
%N A073429 The upper triangular region of the table A073345.
%R A073429
%O A073429 0,6
%K A073429 nonn,tabl,new
%A A073429 Antti Karttunen (my_firstname.my_surname@iki.fi) Jul 31 2002
%D A073429
%p A073429 A073429 := n -> A073345bi(A003056(n),A002262(n));
%p A073429 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
%p A073429 A003056 := n -> floor(sqrt(2*(1+n))-(1/2));


%I A073430
%S A073430 1,1,0,0,2,0,1,0,4,0,0,2,4,8,0,0,2,8,16,16,0,0,0,12,40,48,32,0,1,0,12,80,144,128,64,0,0,2,12,
%T A073430 136,384,448,320,128,0,0,2,12,224,912,1408,1280,768,256,0,0,0,20,328,2048,4096,4544,3456,1792,
%U A073430 512,0,0,2,16,480,4336,11328,14976,13568,8960,4096,1024,0,0,0,20,648,8784,29984,47232,49152
%N A073430 The upper triangular region of the table A073346.
%R A073430
%O A073430 0,5
%K A073430 nonn,tabl,new
%A A073430 Antti Karttunen (my_firstname.my_surname@iki.fi) Jul 31 2002
%D A073430
%p A073430 A073430 := n -> A073346bi(A003056(n),A002262(n));
%p A073430 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
%p A073430 A003056 := n -> floor(sqrt(2*(1+n))-(1/2));



%I A073431
%S A073431 1,1,1,2,3,6,12,28,65,160,408,1074,2898,7998,22508,64426,187251,551730,
%T A073431 1645840,4964876,15130808,46545788,144424944,451715460
%N A073431 Number of separate orbits/cycles to which the gatomorphisms A069767/A069768 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
%F A073431 a(0)=1, a(n) = (1/(2^(n-1))) * Sum_{i=1..(2^(n-1))} (Sum_{j=0..A007814(i)} A073346(n,j)) [An application of the Orbit-counting lemma. See also the conjectural variant implemented as the Maple procedure A073431v2 below].
%R A073431
%O A073431 0,4
%K A073431 nonn,new
%A A073431 Antti Karttunen (my_firstname.my_surname@iki.fi) Jul 31 2002
%D A073431
%Y A073431 Occurs for first time in A073201 as the row 6 (and 8).
%p A073431 A073431 := proc(n) local i,j; if(0 = n) then RETURN(1); else add(add(A073346bi(n,j),j=0..A007814(i)),i=1..(2^(n-1)))/(2^(n-1)); fi; end;
%p A073431 # For some reason this seems to give the same answers (not proved yet):
%p A073431 A073431v2 := proc(n) local i; if(0 = n) then RETURN(1); else (add(A073346bi(n,A007814(i)),i=1..(2^(n-1)))/(2^(n-2)))-1; fi; end;
%p A007814 A007814 := proc(nn) local i,n; n := nn; i := 0; if(0 = n) then RETURN(0); fi; while(0 = (`mod`(n,2))) do n := n/2; i := i+1; od; RETURN(i); end;



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Yours,

Antti Karttunen