<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-8859-1">
<META content="MSHTML 6.00.2600.0" name=GENERATOR></HEAD>
<BODY bottomMargin=0 leftMargin=3 topMargin=0 rightMargin=3>
<DIV>Here is a nice E.G.F.-challenge to all of you
Exceptionally Gifted Folks ... </DIV>
<DIV> </DIV>
<DIV>Valery Liskovets has a nice sequence (along with formula): </DIV>
<DIV><A
href="http://www.research.att.com/projects/OEIS?Anum=A082161">http://www.research.att.com/projects/OEIS?Anum=A082161</A><BR>1,3,16,127,1363,18628,311250,6173791,142190703,3737431895,<BR>110577492346,3641313700916,132214630355700,5251687490704524,<BR>226664506308709858
</DIV>
<DIV> </DIV>
<DIV>I have by chance found a triangle that is related to A082161 </DIV>
<DIV>with a recurrence that is begging for a nice E.G.F. to come along ...
</DIV>
<DIV> </DIV>
<DIV>The triangle is as follows:</DIV>
<DIV>
<DIV>
<DIV>1,<BR>1,1,<BR>1,3,3,<BR>1,7,16,16,<BR>1,15,63,127,127,<BR>1,31,220,728,1363,1363,<BR>1,63,723,3635,10450,18628,18628,<BR>1,127,2296,16836,69086,180854,311250,311250,<BR>1,255,7143,74487,419917,1505041,3683791,6173791,6173791,<BR>1,511,21940,319888,2419473,11449719,37236256,86626584,142190703,142190703,</DIV>
<DIV>...</DIV>
<DIV>and has the simple recurrence: </DIV>
<DIV> </DIV>
<DIV>T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, </DIV>
<DIV>with T(n,0)=1 for n>=0 and T(n,n)=T(n,n-1) for n>0. </DIV>
<DIV> </DIV>
<DIV>EXAMPLE.</DIV>
<DIV>T(5,2) = 220 = 1*1 + 2*15 + 3*63 = 1*T(4,0) + 2*T(4,1) +
3*T(4,2).<BR>T(5,2) = 220 = 31 + 3*63 = T(5,1) + (2+1)*T(4,2).<BR>T(5,3) = 728 =
220 + 4*127 = T(5,2) + (3+1)*T(4,3).<BR> </DIV>
<DIV>I just now submitted this triangle to the OEIS as A102316.</DIV>
<DIV> </DIV>
<DIV>I believe that this triangle should have an interesting E.G.F. ... </DIV>
<DIV>can anyone find one? </DIV>
<DIV> </DIV>
<DIV>Thanks,</DIV>
<DIV> Paul</DIV></DIV></DIV></BODY></HTML>