[seqfan] Re: Number array not found in OEIS
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> Subject: [seqfan] Number array not found in OEIS
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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