[seqfan] Looking for a simpler pattern for that matrix
Gottfried Helms
helms at uni-kassel.de
Mon May 5 20:43:28 CEST 2014
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
More information about the SeqFan
mailing list