querying multidimensional sequences in OEIS

[Jonathan Post]

To give a specific example:

NEW SEQUENCE FROM Jonathan Vos Post

%I A000001
%S A000001 2, 6, 450, 2836181250, 81492043057751910481759423160156250,
4561157026363824997482074305569280581505536351717093893927260661169357729871499327113563125890139588096951624677718591308593750
%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
<http://us.f551.mail.yahoo.com/ym/Compose?To=jvospost2@yahoo.com&YY=14244&y5beta=yes&y5beta=yes&order=down&sort=date&pos=0>),
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 <http://www.research.att.com/%7Enjas/sequences/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.
> > ________________________________________________________________________
> > Check Out the new free AIM(R) Mail -- 2 GB of storage and
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> >
> >
>
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