EDITS OF A080069, A080070, A080675, A106191.

NOTE: In the entry A106191 I have changed Paul Barry's original formula line:
%F A106191 a(n)=sum{k=0..n, binomial(2(n-k), n-k)/(1-2(n-k))}
to a more succinct, but essentially the same formula:
%F A106191 a(n)=sum{k=0..n, binomial(2k, k)/(1-2k)}

(as (n-k) and k run through the same set of values when summing from k=0 to n).
---------------------------------------------------------------


%I A106191
%S A106191 1,1,3,7,17,45,129,393,1251,4111,13835,47427,164999,581023,2066823,
%T A106191 7415703,26805393,97520733,356810313,1312087713,4846614093,17974854933,
%U A106191 66907388973,249872516253,935991743553,3515800038201,13239692841105
%V A106191 1,-1,-3,-7,-17,-45,-129,-393,-1251,-4111,-13835,-47427,-164999,-581023,-2066823,
%W A106191 -7415703,-26805393,-97520733,-356810313,-1312087713,-4846614093,-17974854933,
%X A106191 -66907388973,-249872516253,-935991743553,-3515800038201,-13239692841105
%N A106191 Expansion of sqrt(1-4x)/(1-x).
%C A106191 Row sums of number triangle A106190. Partial sums of A002420.
%C A106191 For n>=1 onward, the absolute values give also the iterates of A122237, starting from 0. (A122237(0), A122237(A122237(0)), A122237(A122237(A122237(0))), ...), this stems from the fact that the sequence gives the positions of terms with binary expansion 1(10){n-1}0 in A014486 (see A080675).
%F A106191 a(n)=sum{k=0..n, binomial(2k, k)/(1-2k)}
%Y A106191 Sequence in context: A018025 A018026 A087953 this_sequence A062810 A113985 A071985
%Y A106191 Adjacent sequences: A106188 A106189 A106190 this_sequence A106192 A106193 A106194
%Y A106191 |a(n)| = A080300(A080675(n)) = A075161(A001348(n)) (for n>=1) = A075163(A000244(A008578(n-2))) = A014137(n-1)+A014138(n-2) = 2*A014137(n-1)-1, for n>=2. (Because binomial(2n+2,n+1)/(2n+1) = 2*A000108(n)) -- Antti Karttunen, Sep 14 2006.
%K A106191 easy,sign
%O A106191 0,3
%A A106191 Paul Barry (pbarry(AT)wit.ie), Apr 24 2005
%E A106191 Barry's formula made more succinct, as well as comments regarding interpretation as absolute values added by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 14 2006.


%I A080675
%S A080675 2,12,52,212,852,3412,13652,54612,218452,873812,3495252,13981012,55924052,
%T A080675 223696212,894784852,3579139412,14316557652,57266230612,229064922452,916259689812,
%U A080675 3665038759252,14660155037012,58640620148052,234562480592212,938249922368852
%N A080675 (5*4^n-8)/6.
%C A080675 These numbers have a simple binary pattern: 10,1100,110100,11010100,1101010100, ... i.e. the nth term has a binary expansion 1(10){n-1}0, that is, there are n-1 10's between the most significant 1 and the least significant 0.
%Y A080675 a(n) = A072197(n-1) - 1 = A014486(|A106191(n)|). a(n) = A079946(A020988(n-2)) for n>=2. Cf. also A122229.
%Y A080675 Sequence in context: A009537 A057547 A043007 this_sequence A007225 A036359 A055703
%Y A080675 Adjacent sequences: A080672 A080673 A080674 this_sequence A080676 A080677 A080678
%K A080675 nonn
%O A080675 1,1
%A A080675 njas, Mar 02 2003
%E A080675 Further comments added by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 14 2006.


%I A080069
%S A080069 0,2,10,44,178,740,2868,11852,47522,190104,735842,3090116,11777124,
%T A080069 48557252,194656036,778669672,3117617996,12677727330,49850271300,
%U A080069 192901051976,795560529352,3243898094388,12977884832332,51055591319170
%N A080069 a(n) = A014486(A080068(n)).
%C A080069 Note that A080068 can be also obtained as iteration of A072795 o A057506.
%H A080069 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/b080069.txt">Table of n, a(n) for n = 0..512</a>
%H A080069 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/a080069.py.txt">Python program for computing this sequence and the associated image.</a>
%H A080069 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/a080069_512.png">Terms a(1)-a(512) drawn as binary strings.</a>
%Y A080069 Same sequence in binary: A080070. Compare to similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245.
%Y A080069 Sequence in context: A005144 A064835 A025590 this_sequence A068551 A099919 A100397
%Y A080069 Adjacent sequences: A080066 A080067 A080068 this_sequence A080070 A080071 A080072
%K A080069 nonn,base
%O A080069 0,2
%A A080069 Antti Karttunen (Firstname.Surname(AT)gmail.com) Jan 27 2003
%E A080069 Python program and Wolfram-like plot added by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 14 2006.

%I A080070
%S A080070 0,10,1010,101100,10110010,1011100100,101100110100,10111001001100,
%T A080070 1011100110100010,101110011010011000,10110011101001100010,
%U A080070 1011110010011011000100,101100111011010001100100
%N A080070 Decimal encoding of parenthesizations produced by simple iteration starting from empty parentheses, and where each successive parenthesization is obtained from the previous by reflecting it as a general tree/parenthesization, then adding an extra stem beneath the root and then reflecting the underlying binary tree.
%C A080070 Corresponding Lisp/Scheme S-expressions are (),(()),(()()),(()(())),(()(())()),(()((())())),(()(())(()())),...
%C A080070 Conjecture: only the terms in positions 0,1,2 and 4 are symmetric, i.e. A057164(A080068(n)) = A080068(n) (equivalently: A036044(A080069(n)) = A080069(n)) only when n is one of {0,1,2,4}. If this is true, then the formula given in A079438 is exact. I (AK) have checked this up to n=404631 with no other occurrence of a symmetric (general) tree.
%H A080070 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/a080070.pdf">Illustration of initial terms</a>
%H A080070 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/a080069.py.txt">Python program for computing this sequence.</a>
%H A080070 A. Karttunen, <a href="http://www.research.att.com/~njas/sequences/a080069_512.png">Terms a(1)-a(512) plotted as a Wolframesque triangle.</a>
%F A080070 a(n) = A007088(A080069(n)) = A063171(A080068(n))
%e A080070 This demonstrates how to get the fourth term 10110010 from the 3rd term 101100. The corresponding binary and general trees plus parenthesizations are shown. The first operation reflects the general tree, the second adds a new stem under the root, and the third reflects the underlying binary tree, which induces changes on the corresponding general tree:
%e A080070 ..............................................
%e A080070 .....\/................\/\/..........\/\/.....
%e A080070 ......\/......\/\/......\/............\/......
%e A080070 .....\/........\/........\/..........\/.......
%e A080070 ......(A057164).(A057548)..(A057163)..........
%e A080070 ........................o.....................
%e A080070 ........................|.....................
%e A080070 ........o.....o.........o...o.........o.......
%e A080070 ........|.....|..........\./..........|.......
%e A080070 ....o...o.....o...o.......o.........o.o.o.....
%e A080070 .....\./.......\./........|..........\|/......
%e A080070 ......*.........*.........*...........*.......
%e A080070 ..[()(())]..[(())()]..[((())())]..[()(())()]..
%e A080070 ...101100....110010....11100100....10110010...
%Y A080070 Compare to similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245. See also A079438, A080067, A080071, A057119.
%Y A080070 Sequence in context: A075171 A106456 A079214 this_sequence A080120 A006937 A037220
%Y A080070 Adjacent sequences: A080067 A080068 A080069 this_sequence A080071 A080072 A080073
%K A080070 base,nonn
%O A080070 0,2
%A A080070 Antti Karttunen (Firstname.Surname(AT)gmail.com) Jan 27 2003
%E A080070 Python program and Wolfram-like plot added by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 14 2006.