I think the example givem without A number, is:<br>

<br>

<table border="0" cellpadding="0" cellspacing="0" width="100%">

<tbody><tr><td align="left" valign="top" width="100"><a href="http://www.research.att.com/%7Enjas/sequences/A007188" title="Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i).">A007188</a></td>

          <td width="5"><br>

</td>

          <td align="left" valign="top">

            Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i).          <br><font size="-1">(Formerly M1722)</font>

          </td>

          <td width="2"><br>

</td>

              <td align="right" valign="top">

            <font size="-2">+0<br>2</font>

          </td>

        </tr>

      </tbody>

</table>

<table cellpadding="2" cellspacing="0">

<tbody><tr><td width="20"><font size="4"><br>

      </font></td><td width="710"><font size="4"><tt>2, 6, 90, 47250, 66852843750, 2806877704512541816406250, 1216935896582703898519354781702537118597533386230468750</tt></font></td></tr></tbody>

</table>

<br>

There might be other Godel-coded sequences, and the like, besides

Conway's brilliant "Fractran", above and beyond those with "decimal

Godelization" -- but usually these quickly lead to numbers too large

for OEIS.<br>

<br>

-- Jonathan Vos Post<br>

<br><div><span class="gmail_quote">On 11/3/06, <b class="gmail_sendername"><a href="mailto:franktaw@netscape.net">franktaw@netscape.net</a></b> <<a href="mailto:franktaw@netscape.net">franktaw@netscape.net</a>> wrote:

</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"><Me><br>>Off hand, I can't think of any interesting functions of finite<br>sequences of

<br>>positive integers that depend on the order of the sequence.<br><br>Of course, I no sooner wrote that than I started thinking of such<br>functions.<br>I'm not sure any of these are tremendously interesting, but they're at

least somewhat so. In the following, b(i), i = 1..k is the finite sequence.  I'm starting all these sequences with the empty sequence, but I searched for them without that first term.  All are shown here in A066099 order.

<br><br>Alternating sum<br>Sum_i (-1)^{i+1) b(i)<br>0,1,2,0,3,1,-1,1,4,2,0,2,-2,0,2,0<br><br>Binomial sum<br>Sum_i C(k-1,i-1) b(i)<br>0,1,2,2,3,3,3,4,4,4,4,5,4,6,5,8<br><br>Inverse binomial sum<br>Sum_i (-1)^{i+1) C(k-1,i-1) b(i)

<br>0,1,2,0,3,1,-1,0,4,2,0,1,-2,-2,1,0<br><br>Weighted sum<br>sum_i i b(i)<br>0,1,2,3,3,4,5,6,4,5,6,7,7,8,9,10<br>This is A029931<br><br>Zero-based weighted sum<br>sum_i (i-1) b(i)<br>0,0,0,1,0,1,2,3,0,1,2,3,3,4,5,6<br><br>

Sum of products of consecutive elements<br>sum_{i=1}^{k-1} b(i) b(i+1)<br>0,0,0,1,0,2,2,2,0,3,4,3,3,4,3,3<br><br>Number of rises<br>sum_{b(i)>b(i-1)} 1<br>0,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0<br><br>Number of falls<br>sum_{b(i)<b(i-1)} 1

<br>0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0<br><br>Number of unchanged<br>sum_{b(i)=b(i-1)} 1<br>0,0,0,1,0,0,0,2,0,0,1,1,0,0,1,3<br><br>Number of non-rises<br>sum_{b(i)<=b(i-1)} 1<br>0,0,0,1,0,0,1,2,0,0,1,1,1,1,2,3<br><br>Number of non-falls

<br>sum_{b(i)>=b(i-1)} 1<br>0,0,0,1,0,1,0,2,0,1,1,2,0,1,1,3<br><br>Number of monotonically increasing runs<br>1 + number of falls, but 0 for empty sequence<br>0,1,1,1,1,2,1,1,1,2,1,2,1,2,1,1<br><br>Number of monotonically decreasing runs

<br>1 + number of rises, but 0 for empty sequence<br>0,1,1,1,1,1,2,1,1,1,1,1,2,2,2,1<br><br>Number of runs of equal terms<br>1 + number of unchanged, but 0 for empty sequence<br>0,1,1,2,1,1,1,3,1,1,2,2,1,1,2,4<br><br>Number of distinct non-empty subsequences

<br>0,1,1,2,1,3,3,3,1,3,2,5,3,5,5,4<br>E.g., f(1,1) = 2, for the sequences [1], and [1,1].<br><br>Number of distinct subsequences<br>1 + number of distinct non-empty subsequences<br>1,2,2,3,2,4,4,4,2,4,3,6,4,6,6,5<br><br>

Number of set partitions<br>List parts of set partition by their smallest element, and count the<br>part sizes.<br>E.g., 1|2,4|3 (also known as {1,2,3,2}) would count for the sequence<br>[1,2,1].<br>1,1,1,1,1,2,1,1,1,3,3,3,1,2,1,1

<br><br>Number of permutations<br>List permutations in cycle form, sorted by the smallest element in the<br>cycle,<br>and count the cycle lengths.<br>Number of set partitions * Product_i (b(i)-1)!<br>1,1,1,1,2,2,1,1,6,6,3,3,2,2,1,1

<br><br>Number of partially ordered sets (unlabelled) by rank<br>The rank of an element in a poset is the length of the longest chain of<br>which it<br>is the largest element.<br>1,1,1,1,1,2,1,1,1,3,4,3,1,2,1,1<br><br>Number of partially ordered sets (labelled) by rank

<br>1,1,1,2,1,9,3,6,1,28,54,60,4,36,12,24<br><br>Number of partially ordered sets (naturally labelled) by rank<br>Naturally labelled means labels are consistent with the ordering.<br>1,1,1,1,1,4,1,1,1,11,13,8,1,4,1,1<br><br>

Number of forests (unlabelled, unordered) with b(i) nodes at height i. 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,2,2,1,1,1,1,2,1,1,1,1,1 ... and, of course, various other forest options. If there is agreement that this is the right order for these sequences,

I'll try to find time to submit them. ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.

<br><br></blockquote></div><br>