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A178904 This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n. 5

%I #16 Mar 16 2020 14:25:59

%S 1,-1,-1,0,-1,0,0,1,1,0,0,-1,1,-1,0,0,1,-1,-1,1,0,0,-1,2,-3,2,-1,0,0,

%T 1,-3,4,4,-3,1,0,0,-1,3,-6,8,-6,3,-1,0,0,1,-3,9,-13,-13,9,-3,1,0,0,-1,

%U 4,-11,19,-23,19,-11,4,-1,0,0,1,-5,13,-27,39,39,-27,13,-5,1,0,0,-1,5,-17,38,-61,71,-61,38,-17,5,-1,0

%N This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.

%C This table is symmetric: a(m,n)=a(n,m) for all m,n>=0.

%H G. C. Greubel, <a href="/A178904/b178904.txt">Table of n, a(n) for the first 100 rows, flattened</a>

%e a(0,0) = 1, a(1,0) = a(0,1) = -1.

%e Triangle begins:

%e 1;

%e -1, -1;

%e 0, -1, 0;

%e 0, 1, 1, 0;

%e 0, -1, 1, -1, 0;

%e 0, 1, -1, -1, 1, 0;

%e 0, -1, 2, -3, 2, -1, 0;

%e ...

%t b[m_, n_] := (-1)^Max[m, n]*Binomial[m+n, n]; A[m_, n_] := DivisorSum[ n+m+1, b[Floor[m/#], Floor[n/#]]*MoebiusMu[#]&]/(m+n+1); Table[A[m-n, n], {m, 0, 12}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Feb 23 2017, adapted from Python *)

%o (Sage)

%o def twisted_binomial(m, n):

%o return (-1)**max(m, n) * binomial(m + n, n)

%o def coefficients_A(m, n):

%o return sum(twisted_binomial(m // d, n // d) * moebius(d)

%o for d in divisors(m + n + 1)) / (m + n + 1)

%o matrix(ZZ, 8, 8, coefficients_A)

%Y Cf. A178738, A178749, A022553, A131868, A163210.

%K sign,tabl

%O 0,24

%A _F. Chapoton_, Jun 22 2010

%E Terms a(82) onward added by _G. C. Greubel_, Dec 10 2017

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Last modified March 29 03:41 EDT 2024. Contains 371264 sequences. (Running on oeis4.)