up and down

[Meeussen Wouter (bkarnd)]

Dear All,

for your consideration (and open to added insights)
-------------
1      1+2+1      1+2+3+2+1     ....  1+2+3+...+n-1+ n +n-1+ ... +3+2+1
equals
1          4                  9                                          n^2
-------------
this is simply P(order=1, n)  == 2*Sum[k ,{k,1,n-1}] +n 
or counting up to n and down again.
-------------
now,  define P(order=2 , m) == 2*Sum[P(1,n) ,{n,1,m-1}] + P(1,m)
meaning :
up & down to n, and this for n from 1 up to m and down again.

Now, P(order=3, __ ) is easy to guess.

Mathematica allows a nesting of pure functions:

updown=NestList[((2*Sum[#,{n,k-1}]+(#/. n->k)//Simplify)/.k->n)&, n, 16]  

  n, 
  n^2,
 1/3 ((n + 2 n^3)),
 1/3 n^2 ((2 + n^2)), 
  1/15 n ((3 + 10 n^2 + 2 n^4)), 
  1/45 n^2 ((23 + 20 n^2 + 2 n^4)), 
  1/315 n ((45 + 196 n^2 + 70 n^4 + 4 n^6)), 
  1/315 n^2 ((132 + 154 n^2 + 28 n^4 + n^6))
    etc...

the denominators are 
ID Number: A049606 Sequence:
1,1,1,3,3,15,45,315,315,2835,14175,155925,467775,6081075,
42567525,638512875,638512875,10854718875,97692469875,
1856156927625,9280784638125,194896477400625,
2143861251406875,49308808782358125,147926426347074375 Name: Denominator of
2^n/n!.

Taking the Coefficient List of     "updown" times A049606        gives a
beautifull table:

{{1},
 {0, 1},
 {1, 0, 2},
 {0, 2, 0, 1}, 
{3, 0, 10, 0, 2},
 {0, 23, 0, 20, 0, 2},
 {45, 0, 196, 0, 70, 0, 4},
 {0, 132, 0, 154, 0, 28, 0, 1},
 {315, 0, 1636, 0, 798, 0, 84, 0, 2},
 {0, 5067, 0, 7180, 0, 1806, 0, 120, 0, 2}, 
{14175, 0, 83754, 0, 50270, 0, 7392, 0, 330, 0, 4},
 {0, 146430, 0, 239327, 0, 74800, 0, 6996, 0, 220, 0, 2},
 {467775, 0, 3056742, 0, 2137564, 0, 393536, 0, 24882, 0, 572, 0, 4},
 {0, 11889315, 0, 21724248, 0, 7971964, 0, 939224, 0, 42042, 0, 728, 0, 4}, 
{42567525, 0, 301942152, 0, 237966820, 0, 51755704, 0, 4142710, 0, 136136,
0, 1820, 0, 8},
 {0, 161368200, 0, 322873524, 0, 134377880, 0, 18796778, 0, 1069640, 0,
26572, 0, 280, 0, 1},
 {638512875, 0, 4851868680, 0, 4215249348, 0, 1045091320, 0, 99734206, 0,
4181320, 0, 80444, 0, 680, 0, 2}}

where the rowsums are A049606  again
and last item of each row appears to be A048896 (funny, that!)

ID Number: A048896 Sequence:
1,1,2,1,2,2,4,1,2,2,4,2,4,4,8,1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,
16,1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,
16,8,16,16,32,1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,
8,16,4,8,8,16,8,16,16,32,2,4,4 Name: Maximal power of 2 dividing n-th
Catalan number

any other observations anyone??

Wouter Meeussen
tel  +32 (0)51 332 124
fax +32 (0)51 332 175
mail: wouter.meeussen at vandemoortele.com




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