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48 Chapter 2. On a conjecture about addition modulo 2 k − 1where ( )li 1,...,i nis a multinomial coefficient and bd,nl,mis defined asn−mb d,nl,m = ∑( n − mii=0) d−n∑( ) d − njj=0⎛⎝ ∑ k≥1∑k j≥0,j∈I∪J,1≤j≤m(l + S − m)!l!2 k [ ] ⎞ h − k ⎠ ∏ A kj∏ A kj − 3 kj=0(h − k)! l + S − m k j ! k j !j∈J j∈Im∏j=1C kj−1|k j − 1|!.Within the above expression for b d,nl,m, the following notation is used:• I = {m + 1, . . . , m + i};• J = {n + 1, . . . , n + j};• S = ∑ j∈I∪J,1≤j≤m k j;• h = d − m − j − i;and⎧⎨ A j + Bj+1j+1if j > 0 ,C j = −⎩13 6if j = 0 ,1 if j = −1 .Here, A i is a sum of Eulerian numbers and B i a Bernoulli number; both are described inSubsection 2.5.3.Finally, we prove in Subsection 2.5.6 an additional property predicted experimentally.Proposition 2.5.3. For 0 < j ≤ i,a d,ni.e. the value of b d,nl,mdoes not depend on m.2.5.1 Experimental resultsFor d = 1, by Theorem 2.4.8, we have(i,j,...) = i + 1 a d,nj(i+1,j−1,...) ;f 1 (β 1 ) = 2 3 4−β1 + 1 3 .The case d = 2 has been treated in Subsection 2.4.5 and leads to a similar expression:f 2 (β 1 , β 2 ) = 11 ( 227 + 4−β1 9 β 1 − 2 )27( 2+ 4 −β2 9 β 2 − 2 ) ( 20+ 4 −β1−β22727 − 2 )9 (β 1 + β 2 ) .In both these cases, f d has the correct form and has been shown to verify Conjecture 1.2.2.The tables in Appendix A give the normalized coefficients a d,n(i 1,...,i n)of the multivariatepolynomials Pd n for the first few d’s. All of these data were computed symbolically using Sage [250],Pynac [268] and Maxima [267]. As a byproduct of this work, the interface between Sage [250]