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<DIV>Seqfans, <BR> Would someone like to try to find a
formula for A112319: <BR>"Coefficients of x^n in the (n-1)-th
self-composition of (x + x^2) for n>=1."</DIV>
<DIV>1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, 1059312264, </DIV>
<DIV> </DIV>
<DIV>The related triangles A122888 and A122890 seem to indicate that </DIV>
<DIV>A112319 may have a formula involving the Catalan numbers: </DIV>
<DIV><A
href="http://www.research.att.com/~njas/sequences/A122890">http://www.research.att.com/~njas/sequences/A122890</A></DIV>
<DIV> </DIV>
<DIV>Below I copy A112319 and triangle A122888 for your convenience. </DIV>
<DIV>Other diagonals in self-compositions of (x+x^2) are A112317, A112320.</DIV>
<DIV> <BR>Be it a recurrence, series reversion, etc., any formula
for these diagonals </DIV>
<DIV>could lead to other discoveries involving self-compositions of
functions. </DIV>
<DIV> Paul </DIV>
<DIV>----------------------------------------------------------------</DIV>
<DIV> </DIV>
<DIV>A122888<BR>Triangle, read by rows, where row n lists the coefficients of
x^k, <BR>k=1..2^n, in the n-th self-composition of (x + x^2) for n>=0. </DIV>
<DIV> </DIV>
<DIV>EXAMPLE </DIV>
<DIV>Triangle begins: <BR>1; <BR>1, 1; <BR>1, 2, 2, 1; <BR>1, 3, 6, 9, 10, 8, 4,
1; <BR>1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1; </DIV>
<DIV> </DIV>
<DIV>Multiplying the g.f. of column k by (1-x)^k, k>=1, with leading zeros,
<BR>yields the g.f. of row k in the triangle A122890: <BR>1; <BR>0, 1; <BR>0, 0,
2; <BR>0, 0, 1, 5; <BR>0, 0, 0, 10, 14; <BR>0, 0, 0, 8, 70, 42; <BR>0, 0, 0, 4,
160, 424, 132; <BR>0, 0, 0, 1, 250, 1978, 2382, 429; <BR>0, 0, 0, 0, 302, 6276,
19508, 12804, 1430; ... </DIV>
<DIV> </DIV>
<DIV>in which the main diagonal is the Catalan numbers <BR>and the row sums form
the factorials. </DIV>
<DIV> </DIV>
<DIV>----------------------------------------------------------------</DIV>
<DIV> </DIV>
<DIV>A112319 <BR>Coefficients of x^n in the (n-1)-th self-composition of
(x + x^2) for n>=1. </DIV>
<DIV> </DIV>
<DIV>1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, 1059312264,
<BR>27715541568, 800423573676, 25289923553700, 867723362137464,
<BR>32128443862364255, 1276818947065793736, 54208515369076658640,
<BR>2448636361058495090816</DIV>
<DIV> </DIV>
<DIV>FORMULA <BR>a(n) = [x^n] F_{n-1}(x) where F_n(x) = F_{n-1}(x+x^2)
<BR>with F_1(x) = x+x^2 and F_0(x)=x for n>=1. <BR> <BR>EXAMPLE
<BR>Initial terms in self-compositions of (x+x^2) are: <BR>F(x) = x + (1)*x^2
<BR>F(F(x)) = x + 2*x^2 + (2)*x^3 + x^4 <BR>F(F(F(x))) = x + 3*x^2 + 6*x^3+
(9)*x^4 +... <BR>F(F(F(F(x)))) = x + 4*x^2 + 12*x^3 + 30*x^4 + (64)*x^5 +...
<BR>F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + (630)*x^6
+...<BR> <BR>----------------------------------------------------------------<BR>END.</DIV></BODY></HTML>