Richard Mathar mathar at strw.leidenuniv.nl
Sat Aug 27 14:30:37 CEST 2011

```http://list.seqfan.eu/pipermail/seqfan/2011-August/015257.html says

gh> From seqfan-bounces at list.seqfan.eu Sat Aug 27 09:32:33 2011
gh> Date: Sat, 27 Aug 2011 00:22:57 +0200
gh> From: Gottfried Helms
gh>
gh>  I came across this number array (should be extended to infinity) by q-brackets/q-binomials
gh>  to base 2 and cannot find a good description. Does someone know this array or has an
gh>  idea how to decode?
gh>
gh>    1    -1       1        -1            1            -1                 1                    -1
gh>   -1     3      -7        15          -31            63              -127                   255
gh>    1    -7      43      -211          931         -3907             16003                -64771
gh>   -1    15    -211      2619       -26251        234795          -1985131              16323819
gh>    1   -31     931    -26251       654811     -13255291         238658491           -4050110011
gh>   -1    63   -3907    234795    -13255291     662827803      -26961325147          973958217435
gh>    1  -127   16003  -1985131    238658491  -26961325147     2699483026843      -220115609012251
gh>   -1   255  -64771  16323819  -4050110011  973958217435  -220115609012251

I guess, up to signs, with offset 0 in both indices, the second column is
T(n,1) = A000225(..)
the third
T(n,2) = 2^(2n+2)-6*2^n+3 = 3-6*2^n+4*4^n.
the fourth
T(n,3) = 2^(3n+3)-28*4^n+42*2^n-21 = -21+42*2^n-28*4^n+8*8^n
the fifth
T(n,4) = 420*4^n-630*2^n-120*8^n+2^(4*n+4)+315
etc. So each column is a sum over powers of 2.
Apparently T(n,k)=T(k,n) (mirror symmetry along the diagonal).

Recurrences down each column appear to take the coefficients in A158474:
T(n,2) =  7*T(n-1,2)-14*T(n-2,2)+  8*T(n-3,2)
T(n,3) = 15*T(n-1,3)-70*T(n-2,3)+120*T(n-3,3)-64*T(n-4,3)
T(n,4) = 31*T(n-1,4)-310*T(n-2,4)+1240*T(n-3,4)-1984*T(n-4,4)+1024 *T(n-5,4)

Richard J. Mathar

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