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<DIV>Seqfans, </DIV>
<DIV> Here are 3 more (compare to (2a) and
(2b) below). </DIV>
<DIV>
<DIV>There seems to be a pattern, but I do not have it all figured out
yet. </DIV>
<DIV>Note that in each of these pairs of problems, such as (4a) and (4b), </DIV>
<DIV>only A is the same; solutions for B,C,D,... are different for each
problem. <BR> Paul </DIV>
<DIV>-------------------------------------------------------------------</DIV>
<DIV>Consider the systems of simultaneous equations below; for each
case, </DIV>
<DIV>what is the unique solution to variable A as a formal power series in
x? <BR>
<DIV>(4a)<BR>A = 1 + xB <BR>B = A + xD<BR>C = B + xF<BR>D = C + xH<BR>E = D +
xJ<BR>...<BR>(4b)<BR>A = 1 + xAB<BR>B = 1 + xBCD<BR>C = 1 + xCDEF<BR>D = 1 +
xDEFGH<BR>E = 1 + xEFGHIJ<BR>...</DIV>
<DIV>(same solution for A as in (4a)).</DIV>
<DIV> <BR>(5a)<BR>A = (1 + xAB)<BR>B = A(1 + xBC)<BR>C = B(1 +
xCD)<BR>D = C(1 + xDE)<BR>E = D(1 + xEF)<BR>...<BR>(5b) <BR>A = 1 +
x(A)^2*B<BR>B = 1 + x(AB)^2*C<BR>C = 1 + x(ABC)^2*D<BR>D = 1 + x(ABCD)^2*E<BR>E
= 1 + x(ABCDE)^2*F <BR>...<BR>(same solution for A as in
(5a)).<BR> </DIV></DIV>
<DIV>(6a)<BR>A = (1 + xC)<BR>B = A(1 + xD)<BR>C = B(1 + xE)<BR>D = C(1 +
xF)<BR>E = D(1 + xG)<BR>...<BR>(6b) <BR>A = 1 + xABC<BR>B = 1 + xABCD<BR>C
= 1 + xABCDE<BR>D = 1 + xABCDEF<BR>E = 1 + xABCDEFG <BR>...<BR>(same solution
for A as in (6a)).</DIV>
<DIV> </DIV>
<DIV> <BR>SOLUTIONS. </DIV>
<DIV>
<DIV>(4a) and (4b):<BR>A = g.f. of A002449 </DIV>
<DIV>(number of different types of binary trees of height n): <BR>[1, 1, 2, 6,
26, 166, 1626, 25510, 664666, 29559718, 2290267226,...]. <BR> </DIV></DIV>
<DIV>(5a) and (5b): <BR>A satisfies: A(x) = 1 + xA(x)^2*A(xA(x)^2) = g.f.
of A088717:<BR>[1, 1, 3, 14, 84, 596, 4785, 42349, 406287, 4176971,
45640572,...]. <BR> </DIV>
<DIV>(6a) and (6b): </DIV>
<DIV>x*A = A(x) satisfies: A(x) = x + x*A(A(A(x))) = g.f. of
A091713, </DIV>
<DIV>and coefficients of A begin:</DIV>
<DIV>[1, 1, 3, 15, 99, 781, 7001, 69253, 742071, 8506775,
103411463,...].<BR> </DIV>
<DIV>END.</DIV>
<DIV> </DIV></DIV>
<DIV>On Mon, 19 Mar 2007 22:55:03 -0400 "Paul D. Hanna" <<A
href="mailto:pauldhanna@juno.com">pauldhanna@juno.com</A>> writes:</DIV>
<BLOCKQUOTE dir=ltr
style="PADDING-LEFT: 10px; MARGIN-LEFT: 10px; BORDER-LEFT: #000000 2px solid">
<DIV></DIV>
<DIV>Consider the systems of simultaneous equations below; for each
case, </DIV>
<DIV>what is the unique solution to variable A as a formal power series
in x? <BR> <BR>(1)<BR>A = 1 + xAB<BR>B = A + xBC<BR>C = B + xCD<BR>D = C
+ xDE<BR>E = D + xEF<BR>...<BR> <BR>(2a) <BR>A = (1 + xB)<BR>B =
A(1 + xC)<BR>C = B(1 + xD)<BR>D = C(1 + xE)<BR>E = D(1 + xF)<BR>...
<BR>(2b) <BR>A = 1 + xAB<BR>B = 1 + xABC<BR>C = 1 + xABCD<BR>D = 1 +
xABCDE<BR>E = 1 + xABCDEF <BR>...<BR>(same solution as
(2a)).<BR> <BR>(3) <BR>A = 1 + xB<BR>B = 1 + xAC<BR>C = 1 + xABD<BR>D = 1
+ xABCE<BR>E = 1 + xABCDF<BR>...</DIV>
<DIV> <BR>SOLUTIONS.<BR>(1) A satisfies: A(x) =
1 + x*A(x)^2*A(xA(x)) = g.f. of A088714: <BR>[1, 1, 3, 13, 69, 419, 2809,
20353, 157199, 1281993, 10963825,...]. <BR> <BR>(2a) and (2b): A
satisfies: A(x) = 1 + xA(x)*A(xA(x)) <BR>where xA(x) = g.f. of A030266,
and coefficients of A begin: <BR>[1, 1, 2, 6, 23, 104, 531, 2982, 18109,
117545, 808764,...]. <BR> <BR>(3) A satisfies: A(x) = 1 + xA(xA(x))
= g.f. of A087949: <BR>[1, 1, 1, 2, 5, 16, 59, 246, 1131, 5655, 30428,
174835,...]. </DIV>
<DIV> </DIV>
<DIV>END.</DIV>
<DIV> </DIV></BLOCKQUOTE></BODY></HTML>