querying multidimensional sequences in OEIS

[franktaw at netscape.net]

This is OK, but it is the opposite of what we started out talking 
about.  This is a function from integers to sequences; but we are 
talking about functions from sequences to integers.  And not every 
integer encodes a sequence in this way; only those in A055932.

Franklin T. Adams-Watters


-----Original Message-----
From: jvospost3 at gmail.com

  To give a specific example:

 NEW SEQUENCE FROM Jonathan Vos Post

%I A000001
%S A000001 2, 6, 450, 2836181250, 81492043057751910481759423160156250,
4561157026363824997482074305569280581505536351717093893927260661169357729
871499327113563125890139588096951624677718591308593750

%N A000001 Multiplicative encoding of Catalan's triangle: Product
p(i+1)^T(n,i).
%C A000001 This is to A009766 "Catalan's triangle T(n,k) (read by
rows)"  as A007188 "Multiplicative encoding of Pascal triangle: Product

p(i+1)^C(n,i)" is to A007318 "Pascal's triangle read by rows."
%F A000001 a(n) = PRODUCT[i=i..n] p(i+1)^T(n,i), where T(n,i), are
Cataln's triangle as in a009766.
%e A000001 a(1) = p(1)^T(1,1) = 2^1 = 2.

a(2) = p(1)^T(2,1) * p(2)^T(2,2) = 2^1 * 3^1 = 6.
a(3) = p(1)^T(3,1) * p(2)^T(3,2) * p(3)^T(3,3) = 2^1 * 3^2 * 5^2 = 450.
a(4) = 2^1 * 3^3 * 5^5 * 7^5 = 2836181250.
a(5) = 2^1 * 3^4 * 5^9 * 7^14 * 11^14 =

81492043057751910481759423160156250.
a(6) = 2^1 * 3^5 * 5^14 * 7^28 * 11^42 * 13^42.
%Y A000001 Cf. A007188, A007318, A009766.
%O A000001 1
%K A000001 ,easy,nonn,
%A A000001 Jonathan Vos Post (
jvospost2 at yahoo.com), Nov 03 2006
RH
RA 192.20.225.32

On 11/3/06, Jonathan Post <jvospost3 at gmail.com> wrote: I think the 
example givem without A number, is:

   A007188
   Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i).
(Formerly M1722)
   +0
2

  2, 6, 90, 47250, 66852843750, 2806877704512541816406250, 
1216935896582703898519354781702537118597533386230468750

  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.

 -- Jonathan Vos Post

On 11/3/06, franktaw at netscape.net < franktaw at netscape.net> wrote: <Me>
>Off hand, I can't think of any interesting functions of finite
 sequences of
>positive integers that depend on the order of the sequence.

Of course, I no sooner wrote that than I started thinking of such
functions.
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.

Alternating sum
Sum_i (-1)^{i+1) b(i)
0,1,2,0,3,1,-1,1,4,2,0,2,-2,0,2,0

Binomial sum
Sum_i C(k-1,i-1) b(i)
0,1,2,2,3,3,3,4,4,4,4,5,4,6,5,8

Inverse binomial sum
Sum_i (-1)^{i+1) C(k-1,i-1) b(i)
0,1,2,0,3,1,-1,0,4,2,0,1,-2,-2,1,0

Weighted sum
sum_i i b(i)
0,1,2,3,3,4,5,6,4,5,6,7,7,8,9,10
This is A029931

Zero-based weighted sum
sum_i (i-1) b(i)
0,0,0,1,0,1,2,3,0,1,2,3,3,4,5,6

 Sum of products of consecutive elements
sum_{i=1}^{k-1} b(i) b(i+1)
0,0,0,1,0,2,2,2,0,3,4,3,3,4,3,3

Number of rises
sum_{b(i)>b(i-1)} 1
0,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0

Number of falls
sum_{b(i)<b(i-1)} 1
0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0

Number of unchanged
sum_{b(i)=b(i-1)} 1
0,0,0,1,0,0,0,2,0,0,1,1,0,0,1,3

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

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

Number of monotonically increasing runs
1 + number of falls, but 0 for empty sequence
0,1,1,1,1,2,1,1,1,2,1,2,1,2,1,1

Number of monotonically decreasing runs
1 + number of rises, but 0 for empty sequence
0,1,1,1,1,1,2,1,1,1,1,1,2,2,2,1

Number of runs of equal terms
1 + number of unchanged, but 0 for empty sequence
0,1,1,2,1,1,1,3,1,1,2,2,1,1,2,4

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

Number of distinct subsequences
1 + number of distinct non-empty subsequences
1,2,2,3,2,4,4,4,2,4,3,6,4,6,6,5

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

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

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

Number of partially ordered sets (labelled) by rank
1,1,1,2,1,9,3,6,1,28,54,60,4,36,12,24

Number of partially ordered sets (naturally labelled) by rank
Naturally labelled means labels are consistent with the ordering.
1,1,1,1,1,4,1,1,1,11,13,8,1,4,1,1

 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.
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