[seqfan] Re: Looking for a simpler pattern for that matrix
Alexander P-sky
apovolot at gmail.com
Tue May 6 07:18:53 CEST 2014
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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