[seqfan] Re: Looking for a simpler pattern for that matrix

[Alexander P-sky]

I noticed that (or rather it appears that) for all rows starting from the
row {-5, 3}

(Abs( the "next to last" term in each row) - 2) / (the last term in the
same row)

yields the integer sequence

 {1,5,11,19,29,41,55,71,89,...}

a(n) = n^2 + n - 1

Where n  >= 1 is the row number (counting rows from n = 0)

PS Further - it appears that this matrix could be described diagonal-wise,
since it appears that every left-to-right diagonal is a sequence, for which
exists a generating function (which could be found empirically, using given
data values).




Alexander R. Povolotsky

On Monday, May 5, 2014, Gottfried Helms <helms at uni-kassel.de> wrote:

> Hi -
>
>  I'm experimenting with some generating functions
>  and by analysis of the resulting approximative values
>  I get the following matrix of coefficients. I'm unable
>  to find a simpler pattern (for instance recursive);
>  possibly there are the Bell-numbers involved because
>  the matrix describes compositions of exponentials.
>
>  Here is the matrix
>  A =
>      2        .          .          .           .          .           .
>       .         .      .      .   .
>     -5        3          .          .           .          .           .
>       .         .      .      .   .
>     10      -22          4          .           .          .           .
>       .         .      .      .   .
>    -17      108        -57          5           .          .           .
>       .         .      .      .   .
>     26     -432        504       -116           6          .           .
>       .         .      .      .   .
>    -37     1520      -3510       1600        -205          7           .
>       .         .      .      .   .
>     50    -4896      21060     -16960        4050       -330           8
>       .         .      .      .   .
>    -65    14784    -113967     152320      -60375       8820        -497
>       9         .      .      .   .
>     82   -42496     571536   -1218560      752500    -175392       17248
>    -712        10      .      .   .
>   -101   117504   -2703132    8945664    -8268750    2884896     -440412
>   31104      -981     11      .   .
>    122  -314880   12203460  -61440000    82687500  -41477184     9219840
> -990720     52650  -1310     12   .
>   -145   822272  -53045685  400097280  -768281250  538876800  -167498562
>  25681920  -2044845  84700  -1705  13
>
> I do not yet know, whether it is meaningful to prefix it with an additional
> column 1,0,0,0,... and also whether the row- and column-indices
> (let's use "r" and "c" for that) better begin at zero or at 1.
>
> It seems, that along the columns, consecutive powers of the column-nr are
> involved, so 1^r in column 1 , 2^r in column 2, 3^r in column 3 and so on.
> For column 1 WolframAlpha gives abs(a_{r,1}) = r^2+1, but for the following
> columns I've not yet an exponential or polynomial formula.
> What I finally want is a formula which allows to compute the columns to
> arbitrary index so that I can approximate the column-sums
> (divergent (but Euler-summable) for columns of A,
>  possibly convergent for columns of F^-1 * A * F )
>
> Thank you in advance for any helpful idea...
>
> Gottfried
>
> P.s.
>
> A bit more context, possibly helpful to suggest an idea:
>
> I get it as precept of coefficients for composition with the columnvector
>
>         E = e * [1 , e, e^2, e^3, ...]         where   e=exp(1)
>
> and the diagonalvector
>
>         F = [0!, 1!, 2!, 3! , ...]
>
>
> by
>         C = (F^-1 * A * F) * E
>
>
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