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A008608
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Number of n X n upper triangular matrices A of nonnegative integers such that a_1i + a_2i + ... + a_{i-1,i} - a_ii - a_{i,i+1} - ... - a_in = -1.
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12
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1, 2, 7, 40, 357, 4820, 96030, 2766572, 113300265, 6499477726, 515564231770, 55908184737696, 8203615387086224, 1613808957720017838, 422045413500096791377, 145606442599303799948900, 65801956684134601408784992, 38698135339344702725297294600, 29437141738828506134939056167071, 28800381656420765181010517468370560
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OFFSET
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1,2
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COMMENTS
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Garsia and Haglund call these Tesler matrices. - N. J. A. Sloane, Jul 04 2014
This is also the value of the type A_n Kostant partition function evaluated at (1,1,...,1,-n) in ZZ^(n+1). This is the number of ways of writing the vector (1,1,...,1,-n) in ZZ^(n+1) as a linear combination with nonnegative integer coefficients of the vectors e_i - e_j, for 1 <= i<j <= n+1. - Alejandro H. Morales, Mar 11 2014
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LINKS
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Michèle Vergne et al., Maple programs for efficient computation of the Kostant partition function.
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EXAMPLE
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For n = 3 there are seven matrices: [[1,0,0],[0,1,0],[0,0,1]], [[1,0,0],[0,0,1],[0,0,2]], [[0,0,1],[0,1,0],[0,0,2]], [[0,0,1],[0,0,1],[0,0,3]], [[0,1,0],[0,2,0],[0,0,1]], [[0,1,0],[0,1,1],[0,0,2]], [[0,1,0],[0,0,2],[0,0,3]], so a(3) = 7. - Alejandro H. Morales, Jul 03 2015
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MAPLE
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multcoeff:=proc(n, f, coeffv, k)
local i, currcoeff;
currcoeff:=f;
for i from 1 to n do
currcoeff:=`if`(coeffv[i]=0, coeff(series(currcoeff, x[i], k), x[i], 0), coeff(series(currcoeff, x[i], k), x[i]^coeffv[i]));
end do;
return currcoeff;
end proc:
F:=n->mul(mul((1-x[i]*x[j]^(-1))^(-1), j=i+1..n), i=1..n):
a := n -> multcoeff(n+1, F(n+1), [seq(1, i=1..n), -n], n+2):
# second Maple program:
b:= proc(n, i, l) option remember; (m-> `if`(m=0, 1,
`if`(i=0, b(l[1]+1, m-1, subsop(1=NULL, l)), add(
b(n-j, i-1, subsop(i=l[i]+j, l)), j=0..n))))(nops(l))
end:
a:= n-> b(1, n-1, [0$(n-1)]):
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MATHEMATICA
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b[n_, i_, l_List] := b[n, i, l] = Function[{m}, If[m==0, 1, If[i==0, b[l[[1]]+1, m-1, ReplacePart[l, 1 -> Sequence[]]], Sum[b[n-j, i-1, ReplacePart[l, i -> l[[i]] + j]], {j, 0, n}]]]][Length[l]]; a[n_] := b[1, n-1, Array[0&, n-1]]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Jul 16 2015, after Alois P. Heinz *)
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CROSSREFS
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Main diagonal (shifted) of A259841.
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KEYWORD
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nonn
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AUTHOR
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Glenn P. Tesler (gptesler(AT)euclid.ucsd.edu)
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EXTENSIONS
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STATUS
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approved
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