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<DIV>If a(n) has (ordinary) generating function A(x), then the generating function for b(n) is B(x) = (A(x)^2 + A(x^2))/2.</DIV>
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<DIV>If we take a(n) to be the Wedderburn-Etherington numbers (A001190), b(n) is the same sequence except that b(1) = 0. This immediately gives rise to the generating function equation for A001190: A(x) = x + (A(x)^2 + A(x^2))/2.</DIV>
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<DIV>Taking a(n)=n+1, b(n) is A005993, Alkane numbers l(6,n). From triangular numbers (n+1)*(n+2)/2, we get A005995, Alkane numbers l(8,n).</DIV>
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<DIV>Taking a(n)=P(n-1)-1=A000065(n-1), the resulting sequence is A000147, trees of diameter 5. Similarly, starting with rooted trees of height 3 (A000235) will produce trees of diameter 7 (A000550), etc.</DIV>
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<DIV>Finally, I just submitted:</DIV>
<DIV><FONT face="Courier New">%I A000001<BR>%S A000001 1,1,2,3,7,14,35,85,226,600,1658,4622,13141,37699,109419,320017,943329,2797788,<BR>8346030,25019401,75340824,227777899,691146578,2104028507,6424449318,19670277332,<BR>60378290912,185763773723,572764664975,1769533823103,5477077872168,16982128750507<BR>%N A000001 Number of binary trees of weight n where leaves have positive integer <BR>weights, where the order of subtrees is insignificant. Commutative <BR>non-associative version of partitions of n.<BR>%F A000001 a(2n) = 1 + C(a(n)+1,2) + sum_{k=1}^{n-1} a(k)*a(2n-k). a(2n+1) = <BR>1 + sum_{k=1}^n a(k)*a(2n+1-k), with a(0)=0.<BR>%e A000001 For a(4)=7, we have the following 7 sums: 4, 3+1, 2+2, (2+1)+1, <BR>(1+1)+2, ((1+1)+1)+1, (1+1)+(1+1).<BR>%Y A000001 Cf A007317, A000041.<BR>%O A000001 0<BR>%K A000001 ,easy,nonn,<BR>%A A000001 Frank Adams-Watters (</FONT><A href="javascript:parent.ComposeTo('FrankTAW%40Netscape.net');"><FONT face="Courier New">FrankTAW@Netscape.net<!
/FONT></A><FONT face="Courier New">), Jan 23 2006<BR></FONT></DIV>
<DIV><FONT face="Courier New">To which I should add</FONT></DIV>
<DIV><FONT face="Courier New"></FONT> </DIV>
<DIV><FONT face="Courier New">%F A000001 G.f. A(x) = ((A(x)-1)^2 + A(x^2) - 1)/2 + 1/(1-x).</DIV></FONT>
<DIV>Franklin T. Adams-Watters<BR>16 W. Michigan Ave.<BR>Palatine, IL 60067<BR>847-776-7645</DIV>
<DIV> </DIV> <BR>-----Original Message-----<BR>
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<DIV>I have recently encountered a transformation several times that I have never seen described. It basically takes a sequence a, where a(n) is the number of objects (of some type) of size n, and produces the sequence of the number of unordered pairs of such objects. ...</DIV></DIV></DIV></DIV></DIV></DIV>
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