R_r recurrence even simpler than S_r

[Gottfried Helms]

Am 04.04.04 01:39 schrieb Rainer Rosenthal:

> Dear SeqFan,
> 
> 
> We may consider (1) as the special case of 
> 
>       a(n-1)*a(n+1) = a(n)^2 - 1         (2)
> 
> where a(n) = n = 0, 1, 2, ...
> 
> Looking for other sequences 0, 1, r, ... obeying (2) we
> quickly find many of them already in the OEIS:
> 
> r= 2: 0,1,2,3,4,5,6,...      "the numbers"
> r= 3: 0,1,3,8,21,55,144,...  A001906
> r= 4: 0,1,4,15,56,209,...    A001353
> r= 5: 0,1,5,24,115,551,...   A004254
> r= 6: 0,1,6,35,204,1189,...  A001109
> 
> followed by A004187, A001090, A018913, A004189, A004190,
> A004191, A078362, A007655, A078364, A077412, A078366,
> A049660, A078368, A075843 (r=7 up to r=20).
> 

This list of sequences made me think of the analysis of the
entries of a *transposed* A_x(n) for another simple scheme.
Also this view suggests the simple formulation of just 2 parameters, see below.


             (r=1)       r=2       r=3       r=4       r=5       r=6       r=7       r=8
Q_b     |         "the numbers" A001906   A001353   A004254   A001109   A004187   A001090
--------+--------------------------------------------------------------------------------+--
        | n=     0         1         2         3         4         5         6         7
--------+--------------------------------------------------------------------------------+--
Q-1(n)  |        0         0         0         0         0         0         0         0
Q0(n)  	|        1         1         1         1         1         1         1         1 ...
Q1(n)  	|        1         2         3         4         5         6         7         8 ...
Q2(n)  	|        0         3         8        15        24        35        48        63 ...
Q3(n)  	|       -1         4        21        56       115       204       329       496 ...
Q4(n)  	|        0         5        55       209       551      1189      2255      3905 ...
Q5(n)  	|        1         6       144       780      2640      6930     15456     30744 ...
Q6(n)  	|        0         7       377      2911     12649     40391    105937    242047 ...
Q7(n)  	|       -1         8       987     10864     60605    235416    726103   1905632 ...
Q8(n)  	|        0         9      2584     40545    290376   1372105   4976784  15003009 ...


All sequences Q_b(n) can be described by this little scheme:

Q0(n) =   1
Q1(n) =   n
Q2(n) =   n²-1
Q3(n) =  (n²-2)n
Q4(n) =  (n²-3)n²+ 1
Q5(n) = ((n²-4)n²+ 3)n
Q6(n) = ((n²-5)n²+ 6)n²-1
Q7(n) =(((n²-6)n²+10)n²-4)n

which can also be continued to negative n, and I would like to see, whether
we can continue to negative b, like, for instance in the Euler-triangle.

The generation rule of the fomulas itself is obvious and simple. Another approach
is using the derivatives, to avoid the various explicit binomials.
(The single-quote marks the derivative dQ_b(n)/dn, and for higher derivatives the {})


          b     b-1 '      1   b-2 ''     1   b-3 '''      1   b-4 {4}              1  b-b {b}
Q_b(n) = n   - n        + --- n        - --- n         + ---- n        + ... - ... ---n
                           2!            3!                4!                       b!

            b    [    1   b-i (i)      i  ]
       =   sum   [  ---- n         *(-1)  ]
           i=0   [    i!                  ]

Perhaps a more concise interpretation.

It is really astonishing, after Rainer Rosenthal found all these Axxxx-sequences,
that they obviously were collected in different fields, so that the relatively
simple recursion-scheme was not of interest earlier. Now, that made me curious
about these fields...

Regards-

Gottfried Helms