A214808 - OEIS

%I #18 May 27 2024 22:57:32

%S 1,1,3,26,898,128826,82462230,244868962698,3481404811793685,

%T 242641164238940173212

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

%C 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

%H MathOverflow, <a href="http://mathoverflow.net/questions/103079">Kostant partition function: asymptotics and specifics</a>.

%F See MathOverflow link for a recurrence.

%e For n = 2, the a(2) = 3 solutions are

%e   0*(1, -1, 0) + 2*(1, 0, -1) + 0*(0, 1, -1) = (2, 0, -2),

%e   1*(1, -1, 0) + 1*(1, 0, -1) + 1*(0, 1, -1) = (2, 0, -2),

%e   2*(1, -1, 0) + 0*(1, 0, -1) + 2*(0, 1, -1) = (2, 0, -2).

%p multcoeff:=proc(n, f, coeffv, k)

%p    local i, currcoeff;

%p    currcoeff:=f;

%p      for i from 1 to n do

%p         currcoeff:=`if`(coeffv[i]=0, coeff(series(currcoeff, x[i], k), x[i], 0), coeff(series(currcoeff, x[i], k), x[i]^coeffv[i]));

%p      end do;

%p      return currcoeff;end proc:F:=n->mul(mul((1-x[i]*x[j]^(-1))^(-1), j=i+1..n), i=1..n):

%p a := n -> multcoeff(n+1, F(n+1), [seq(n-2*i, i=0..n)], 2*n):

%p seq(a(i), i=0..5) # _Alejandro H. Morales_, May 24 2024

%K nonn,more

%O 0,3

%A _N. J. A. Sloane_, Jul 30 2012

%E a(0) and a(7)-a(9) from _Alejandro H. Morales_, May 24 2024