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<DIV> Given two sequences with known g.f., what is the
g.f. of the sequence </DIV>
<DIV>resulting from the term-by-term product of the sequences?</DIV>
<DIV> </DIV>
<DIV>Suppose we call this the "BIPRODUCT" of the two sequences </DIV>
<DIV>(is there a better term for this?).</DIV>
<DIV> </DIV>
<DIV>Then, so far I have derived: </DIV>
<DIV>(1) </DIV>
<DIV>BIPRODUCT( 1/(1-ax-bx^2), 1/(1-cx-dx^2) ) =<BR>(1 - bd*x^2)/(1 - ac*x -
(da^2 + bc^2 + 2bd)*x^2 - abcd*x^3 + b^2*d^2*x^4)</DIV>
<DIV> </DIV>
<DIV>EXAMPLE:</DIV>
<DIV>the biproduct of Fibonacci and Pell is A001582:</DIV>
<DIV>{1*1, 1*2, 2*5, 3*12, 5*29, 8*70, 13*169, ...} =</DIV>
<DIV>{1,2,10,36,145,560,2197,8568,33490,130790,510949,1995840,7796413,30454814,...}</DIV>
<DIV>where the G.f. is given by: </DIV>
<DIV> </DIV>
<DIV>BIPRODUCT( 1/(1-x-x^2), 1/(1-2*x-x^2) ) = </DIV>
<DIV> (1-x^2)/(1-2*x -7*x^2 -2*x^3 +x^4)</DIV>
<DIV> </DIV>
<DIV>Q: Can someone extend this to higher orders --</DIV>
<DIV>i.e., what is biproduct( 1/(1-ax-bx^2-cx^3), 1/(1-dx-ex^2-fx^3) ) = ?</DIV>
<DIV>----------------------------------------------------------------------</DIV>
<DIV> </DIV>
<DIV>(2) Catalan biproducts with 2nd order recurrence.</DIV>
<DIV>Given the recurrence: </DIV>
<DIV> S(n) = b*S(n-1) + c*S(n-2), <BR>then the g.f. for
the term-by-term product: </DIV>
<DIV> a(n) = S(n)*Catalan(n) <BR>is given by: </DIV>
<DIV> sum_{n>=0} S(n)*Catalan(n)*x^n </DIV>
<DIV> </DIV>
<DIV>= sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c)
)/x.<BR> <BR>= (1/x)* series reversion of x*( sqrt(1-4*c*x^2) - b*x
)</DIV>
<DIV> </DIV>
<DIV>EXAMPLE:</DIV>
<DIV>Biproduct of Catalan and Fibonacci (A098614):</DIV>
<DIV>{1,1,4,15,70,336,1716,9009,48620,267410,1494844,8465184,...}</DIV>
<DIV>has g.f.:</DIV>
<DIV>BIPRODUCT( G000108(x), G000045(x) ) =</DIV>
<DIV> sqrt( (1-2*x - sqrt(1-4*x-16*x^2))/10 )/x.</DIV>
<DIV> </DIV>
<DIV>
<DIV>So the BIPRODUCT of two sequences retains some of the properties of
both.</DIV>
<DIV>------------------------------------------------------------------------</DIV></DIV>
<DIV> </DIV>
<DIV>Would this interest anyone to come up with a more general formula?</DIV>
<DIV> </DIV>
<DIV>Thanks,</DIV>
<DIV> Paul</DIV></BODY></HTML>