Cheers Neil,

CC: Henry (concerning the table A073346 and the attachment A073345.txt
which I further edited)
and Wouter (concerning the necklace/bracelet sequences
which have recently popped up, and some of whose entries
I have cleaned up here).
Please object if I have made a mistake somewhere...


Here comes an addition to A000096, re-edits of six sequences,
and three pre-numbered new sequences:
A074079 & A074080 & A074092.
plus also the new version of
http://www.research.att.com/~njas/sequences/A073345.txt
included as an attachment.


First, add the %Y-line to A000096:

%I A000096 M1356 N0522
%S A000096 0,2,5,9,14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,
%Y A000096 Occurs as a diagonal in A074079/A074080, i.e.: A074079(n+3,n) = A000096(n-1) for all n >= 2. Also A074092(n) = 2^n * A000096(n-1) after n >= 2.



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

Complete re-edits of A007595, A007123, A072506, A073201, A073346 and A073431 :

In A007595 I took the Cameron's description
from the list given in
http://www.math.uwaterloo.ca/JIS/VOL3/groupdata.html
(which matches what I had in mind), and changed the
old %N-line to be %F-line.
I also added my name after Wouter's (the %C-line concerning
the necklace interpretation), as I understand that Wouter
just observed the similarity, which I then proved on my
12th August message "Necklaces of binary trees" on SeqFan-list,
but you can take that off, if Wouter has sent a proof of
his own to you in private mail.


%I A007595 M2681
%S A007595 1,1,3,7,22,66,217,715,2438,8398,29414,104006,371516,1337220,
%T A007595 4847637,17678835,64823110,238819350,883634026,3282060210,
%U A007595 12233141908,45741281820,171529836218,644952073662,2430973304732
%N A007595 Rooted planar binary trees upto reflection (of n internal nodes = total 2n+1 nodes).
%F A007595 a(0) = 1 (not explicitly listed here), a(n) = C_n / 2 if n is even or ( C_n + C_((n-1)/2) ) / 2 if n is odd, C = Catalan numbers (A000108).
%C A007595 Also number of necklaces of 2 colours with 2n beads and n-1 black ones. - Wouter Meeussen (wouter.meeussen@pandora.be), Aug 03 2002 and Antti Karttunen, Aug 12 2002.
%D A007595 P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183.
%H A007595 P. J. Cameron, <a href="http://www.math.uwaterloo.ca/JIS/index.html#P00.1.5">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%t A007595 Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#,(n-1)/# ] &)/@Intersection[Divisors[2n],Divisors[n-1]])/(2n),{n,2,32}] or Table[If[EvenQ[n],cat[n]/2,(cat[n] +cat[(n-1)/2])/2],{n,24}] with cat[n]=A000108
%Y A007595 A007595(n)=A047996(2*n,n-1) for n>= 1, and A007595(n)=A072506(n,n-1) for n>=2. Occurs in A073201 as the rows 0, 2, 4, etc (with a(0)=1 included). Cf. also A003444, A007123.
%K A007595 nonn,easy,new
%O A007595 1,3
%p A007595 A007595 := n -> (1/2)*(Cat(n) + (`mod`(n,2)*Cat((n-1)/2)));
%p A007595 Cat := n -> binomial(2*n,n)/(n+1);
%A A007595 njas
%E A007595 Description corrected by Reiner Martin and Wouter Meeussen, Aug 04 2002. Further notes and Maple code from Antti Karttunen (my_firstname.my_surname@iki.fi), Aug 19 2002


%I A007123 M1218
%S A007123 1,1,2,4,10,26,76,232,750,2494,8524,29624,104468,372308,1338936,
%T A007123 4850640,17685270,64834550,238843660,883677784,3282152588,
%U A007123 12233309868,45741634536,171530482864,644953425740,2430975800876
%N A007123 Connected unit interval graphs with n nodes; also bracelets (turn over necklaces) with n black beads and n-1 white beads.
%C A007123 Also rooted planar general trees (of n vertices or n-1 edges) upto reflection, - AK, Aug 09, 2002 (for the correspondence with bracelets, start by considering Raney's lemma as explained by Graham, Knuth & Patashnik).
%D A007123 R. W. Robinson, personal communication.
%D A007123 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 345 & 346.
%H A007123 P. J. Cameron, <a href="http://www.math.uwaterloo.ca/JIS/index.html#P00.1.5">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A007123 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/NecklaceInfo.html">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A007123 <a href="http://www.research.att.com/~njas/sequences/Sindx_Br.html#bracelets">Index entries for sequences related to bracelets</a>
%F A007123 a(n) = (Cat(n)+binomial(n,floor(n/2)))/2 = (A000108(n)+A001405(n))/2. - Antti Karttunen, Aug 09, 2002
%Y A007123 Occurs as the row 164 in A073201. Cf. also A007595, A003239.
%Y A007123 The next-to-centre columns of triangle A052307.
%K A007123 nonn,nice
%O A007123 1,3
%A A007123 njas
%E A007123 Extended by Christian G. Bower (bowerc@usa.net), additional notes from AK, Aug 19 2002.


%I A072506
%S A072506 1,1,2,1,3,4,1,4,7,10,1,5,12,22,26,1,6,19,43,66,80,1,7,26,73,143,217,
%T A072506 246,1,8,35,116,273,504,715,810,1,9,46,172,476,1038,1768,2438,2704,1,
%U A072506 10,57,245,776,1944,3876,6310,8398,9252,1,11,70,335,1197,3399,7752
%N A072506 Triangle giving T(n,m) = number of necklaces of two colours with 2n beads of which m=1..n are black.
%C A072506 Left half of even rows of triangle A047996 (with the leftmost edge discarded).
%F A072506 (1/(2n)) Sum_{d |(2n,m)} phi(d)*binomial(2n/d,m/d)
%t A072506 Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#,m/# ] &)/@Intersection[Divisors[2n], Divisors[m]])/(2n),{n,13},{m,n}]
%Y A072506 Penultimate entries give binary necklaces of n-1 black beads and n+1 white beads: A007595(n)=A072506(n,n-1) for n>=2; antepenultimate entries give binary necklaces of n-2 black beads and n+2 white beads, presumably A003444.
%Y A072506 Cf. A047996, A003444.
%K A072506 nonn,new,tabl
%O A072506 1,3
%A A072506 Wouter Meeussen (wouter.meeussen@pandora.be), Aug 03 2002


%I A073201
%S A073201 1,1,1,1,1,1,3,2,1,1,7,4,1,1,1,22,11,3,1,1,1,66,31,7,2,1,1,1,217,96,22,
%T A073201 4,3,1,1,1,715,305,66,11,7,2,1,1,1,2438,1007,217,30,22,4,2,2,1,1,8398,
%U A073201 3389,715,93,66,11,3,5,1,1,1,29414,11636,2438,292,217,30,6,14,2,2,1,1
%N A073201 Array of cycle count sequences for the table A073200.
%C A073201 Each row of this table gives the counts of separate orbits/cycles to which the gatomorphism given in the corresponding row of A073200 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
%C A073201 Note that for involutions (self-inverse gatomorphisms) this is always (A000108(n)+Affffff(n))/2, where Affffff is the corresponding "fix-count sequence" from the table A073202.
%H A073201 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(With the complete source and explanation)</I>
%Y A073201 Few EIS-sequences which occur in this table. Only the first known occurrence(s) given (marked with ? if not yet proven/unclear):
%Y A073201 Rows 0, 2, 4, etc.: A007595, Row 1: A073191, Rows 6 (& 8): A073431, Row 7: A000108, Rows 12, 14, 20, ...: A057513, Rows 16, 18, ...: A003239, Row 57, ..., 164: A007123, Row 168: A073193, Row 261: A002995, Row 2614: A057507, Row 2618 (?), row 17517: A001683.
%K A073201 nonn,tabl
%O A073201 0,7
%A A073201 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002



%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,
%T A073346 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,
%U A073346 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
%N 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.
%C A073346 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 A073346 ____________________\/\/\/\/_
%C A073346 _____________\/__\/__\/__\/__
%C A073346 ______________\__/____\_ /___
%C A073346 ____.____\/____\/______\/____ etc.
%C A073346 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.
%H A073346 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 A073346 (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?)
%e A073346 The top-left corner of this square array:
%e A073346 1 1 0 1 0 0 0 1 ...
%e A073346 0 0 2 0 2 2 0 0 ...
%e A073346 0 0 0 4 4 8 12 12 ...
%e A073346 0 0 0 0 8 16 40 80 ...
%p A073346 A073346 := n -> A073346bi(A025581(n),A002262(n));
%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);
%Y A073346 Variant: A073345. The first row: A036987. Column sums: A000108. Diagonals: A073346(n,n) = A000007(n), A073346(n+1,n) = A000079(n), A073346(n+2, n) = A058922(n), A073346(n+3,n) = A074092(n) - [see the attached notes.]
%Y A073346 A073430 gives the upper triangular region of this array. Used to compute A073431. Entries on the row k are all divisible by 2^k, thus dividing them out yields the array/triangle A074079/A074080.
%K A073346 nonn,tabl,new
%O A073346 0,8
%A A073346 Antti Karttunen (my_firstname.my_surname@iki.fi) Jul 31 2002


%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)) = (1/(2^(n-2))) * Sum_{i=1..(2^(n-1))} A073346(n,A007814(i)) - 1  = (1/2^n) * Sum_{i=0..n} (2^(n-i))*A073346(n,i) = Sum_{i=0..n} A074079(n,i)
%p A073431 A073431 := proc(n) local i; (1/2^n) * add((2^(n-i))*A073346bi(n,i),i=0..n); end;
%Y A073431 Occurs for first time in A073201 as row 6 (and 8). Column sums of the square array A074079/Row sums of the triangle A074080.
%K A073431 nonn,new
%O A073431 0,4
%A A073431 Antti Karttunen (my_firstname.my_surname@iki.fi) Jul 31 2002


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



%I A074079
%S A074079 1,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,0,2,1,0,0,0,0,0,0,3,2,
%T A074079 0,0,0,0,0,0,1,3,5,1,0,0,0,0,0,0,1,3,10,3,0,0,0,0,0,0,0,0,3,17,9,1,0,0,0,0,0,0,0,1,
%U A074079 5,28,24,4,0,0,0,0,0,0,0,0,0,4,41,57,14,1,0,0,0,0,0,0,0
%N A074079 Square array A(row,col) (listed in order A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), etc), giving essentially the same information as the triangle A074080 which shows only the upper triangular region.
%F A074079 A074079(n,k) = A073346(n,k)/(2^k)
%R A074079
%O A074079 0,31
%K A074079 nonn,tabl,new
%Y A074079 Obtained from the square array A073346 by dividing the entries on the kth row by 2^k. Column sums: A073431. See A074080 for explanation. Cf. also A025581, A002262.
%A A074079 Antti Karttunen (my_firstname.my_surname@iki.fi) Aug 19 2002
%D A074079
%p A074079 A074079bi := (n,k) -> A073346bi(n,k)/(2^k);
%p A074079 A074079 := n -> A074079bi(A025581(n),A002262(n));
%p A074079 A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);
%p A074079 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);




%I A074080
%S A074080 1,0,1,1,0,1,0,1,1,1,0,1,2,2,1,0,0,3,5,3,1,1,0,3,10,9,4,1,0,1,3,17,24,14,5,1,0,1,3,28,
%T A074080 57,44,20,6,1,0,0,5,41,128,128,71,27,7,1,0,1,4,60,271,354,234,106,35,8,1,0,0,5,81,549,937,738,
%U A074080 384,150,44,9,1,0,0,5,106,1061,2396,2247,1326,588,204,54,10,1,0,0,6,136,1973,5927,6650,4430
%N A074080 Triangle T(n,k) (listed in order T(1,0), T(2,0), T(2,1), T(3,0), T(3,1), T(3,2), T(4,0), etc.) giving the number of 2^k -cycles that occur in the nth sub-permutation of A069767/A069768 (i.e. in the range [A014137(n-1)..A014138(n-1)] inclusive).
%e A074080 If we take the fifth such sub-permutation, i.e. the subsequence A069767[23..64]: [45,46,48,49,50,54,55,57,58,59,61,62,63,64,44,47,53,56,60,43,52,40,31,32,41,34,35,36,42,51,39,30,33,38,29,26,27,37,28,25,24,23], subtract 22 from each term, and convert the resulting permutation of [1..42] to disjoint cycle notation, we get:
%e A074080 (17,31),(20,21,30,29),(3,26,12,40),(6,32,8,35,7,33,11,39),(15,22,18,34,16,25,19,38),(1,23,9,36,4,27,13,41,2,24,10,37,5,28,14,42)
%e A074080 which implies that T(5,0) = 0 (no fixed elements), T(5,1) = 1 (one transposition), T(5,2) = 2 (two 4-cycles), T(5,3) = 2 (two 8-cycles), T(5,4) = 1 (and one 16-cycle). It is guaranteed that only cycles whose length is a power of 2 occur in A069767/A069768.
%R A074080
%O A074080 0,13
%K A074080 nonn,tabl,new
%A A074080 Antti Karttunen (my_firstname.my_surname@iki.fi) Aug 19 2002
%Y A074080 Upper triangular region of the square array A074079 (actually, only the area above its main diagonal, excluding also the leftmost column). A074080(n,k) = A073430(n,k)/(2^k) [with the rightmost edge of A073430 discarded]. Row sums: A073431. A000108(n) = Sum_{i=0..n-1} 2^i * A074080(n,i). Cf. A073346, A003056, A002262.
%D A074080
%p A074080 A074079bi := (n,k) -> A073346bi(n,k)/(2^k);
%p A074080 A074080 := n -> A074079bi(A003056(n)+1,A002262(n));
%p A074080 A003056 := n -> floor(sqrt(2*(1+n))-(1/2));
%p A074080 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);



%I A074092
%S A074092 1,2,8,40,144,448,1280,3456,8960,22528,55296,133120,315392,737280,1703936,3899392,8847360,19922944,
%T A074092 44564480,99090432,219152384,482344960,1056964608,2306867200,5016387584,10871635968,23488102400,
%U A074092 50600083456,108716359680,233001975808,498216206336,1063004405760,2263447764992,4810363371520
%N A074092 Number of plane binary trees of size n+3 and contracted height n.
%F A074092 a(0) = 1, a(1) = 2, a(n) = 2^(n-1)*(n+2)*(n-1) = (2^n)*(C(n,n-2)+C(n-1,n-2)) = 2^n * A000096(n-1).
%R A074092
%H A074092 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>
%O A074092 0,2
%K A074092 nonn,new
%A A074092 Antti Karttunen (my_firstname.my_surname@iki.fi) Aug 19 2002
%Y A074092 A074092(n) = A073346(n+3,n).
%D A074092
%p A074092 A074092 := n -> `if`((n < 2),n+1,2^(n-1)*(n+2)*(n-1));
%p A074092 A074092v2 := n -> `if`((n < 2),n+1,(2^n)*(binomial(n,n-2)+binomial(n-1,n-2)));



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


Yours,

Antti Karttunen

BTW: the gatomorphisms A069767 (and it its inverse A069768)
have an interesting property when it is restricted to
the subset of (n,n)-binary trees (i.e. the plane binary trees
of size n and height n): If one maps these to 2^(n-1)
binary strings, such that one starts from the root of the
tree, and from the lsb-end of the binary string, such
that whenever the next higher \/ -node is at the left
hand side, one marks 1, and when it is at the right hand
side, one marks 0, thus for example the binary tree

  \/
 \/
\/
 \/
\/

maps to 0., 01., 010., 0100, reversed: 0010 = 2 in decimal.
The other of these induces an increment by one on these binary words
and the inverse correspondingly the decrement by one!

The attachment for the new version of
http://www.research.att.com/~njas/sequences/A073345.txt
follows: