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A214808
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Kostant partition function P(2*rho) for root system A_n, where 2*rho is the sum of the positive roots.
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0
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OFFSET
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0,3
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COMMENTS
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This is the number of ways of writing the vector 2*rho = (n,n-2,...,-n+2,-n) as a sum of the vectors e_i - e_j for 1 <= i < j <= n+1 with nonnegative integer coefficients. - Alejandro H. Morales, May 24 2024
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LINKS
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FORMULA
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See MathOverflow link for a recurrence.
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EXAMPLE
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For n = 2, the a(2) = 3 solutions are
0*(1, -1, 0) + 2*(1, 0, -1) + 0*(0, 1, -1) = (2, 0, -2),
1*(1, -1, 0) + 1*(1, 0, -1) + 1*(0, 1, -1) = (2, 0, -2),
2*(1, -1, 0) + 0*(1, 0, -1) + 2*(0, 1, -1) = (2, 0, -2).
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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(n-2*i, i=0..n)], 2*n):
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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