A214808 Kostant partition function P(2*rho) for root system A_n, where 2*rho is the sum of the positive roots.

1, 1, 3, 26, 898, 128826, 82462230, 244868962698, 3481404811793685, 242641164238940173212

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..9.

MathOverflow, Kostant partition function: asymptotics and specifics.

Formula

See MathOverflow link for a recurrence.

Example

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).

Maple

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):
seq(a(i), i=0..5) # Alejandro H. Morales, May 24 2024

Crossrefs

Sequence in context: A280222 A354652 A174811 * A378063 A219895 A088730

Adjacent sequences: A214805 A214806 A214807 * A214809 A214810 A214811

Keyword

nonn,more

Author

N. J. A. Sloane, Jul 30 2012

Extensions

a(0) and a(7)-a(9) from Alejandro H. Morales, May 24 2024

Status

approved