[seqfan] Re: Looking for a simpler pattern for that matrix
franktaw at netscape.net
franktaw at netscape.net
Mon May 5 21:02:29 CEST 2014
It looks like it might be related to Stirling numbers of the 2nd kind.
Franklin T. Adams-Watters
-----Original Message-----
From: Gottfried Helms <helms at uni-kassel.de>
To: M <SeqFanList> <seqfan at list.seqfan.eu>
Sent: Mon, May 5, 2014 1:43 pm
Subject: [seqfan] Looking for a simpler pattern for that matrix
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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