%I #20 May 11 2024 19:11:44

%S 0,0,0,1,16,126,672,2772,9504,28314,75504,184041,416416,884884,

%T 1782144,3426384,6325632,11267532,19442016,32605881,53300016,85131970,

%U 133138720,204246900,307850400,456528150,666928080,960846705,1366537536,1920285576,2668289536

%N a(n) = (1/(1!*2!*3!*4!))*Sum_{1 <= x_1, x_2, x_3, x_4 <= n} |det V(x_1,x_2,x_3,x_4)|, where V(x_1,x_2,x_3,x_4) is the Vandermonde matrix of order 4.

%C Compare with A000292 and A040977 for the corresponding sums for the Vandermonde matrices of orders 2 and 3 respectively.

%C a(n) = sum of dimensions of all irreducible polynomial representations of GL(4) whose highest weight is of the form (m1 >= m2 >= m3 >= m4) and m1 <= n. - Oded Yacobi (oyacobi(AT)math.ucsd.edu), Jul 24 2008

%H T. D. Noe, <a href="/A133111/b133111.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = (1/288)*Sum_{1 <= i,j,k,l <= n} |(i-j)(i-k)(j-k)(i-l)(j-l)(k-l)|.

%F G.f.: x^4*(1 + 5*x + 5*x^2 + x^3)/(1 - x)^11 .

%F a(n) = n^2*(n^2 - 1)^2*(n^2 - 4)*(n^2 - 9)/302400.

%F a(n) = Sum_{i + j + k + l = n} i*j*k^2*l^3.

%t a[n_] := n^2 (n^2 - 1)^2 (n^2 - 4) (n^2 - 9)/302400; Array[a, 30] (* _Robert G. Wilson v_, Sep 17 2007 *)

%t Rest@ CoefficientList[ Series[x^4*(1 + 5 x + 5 x^2 + x^3)/(1 - x)^11, {x, 0, 30}], x] (* _Robert G. Wilson v_, Sep 17 2007 *)

%Y Cf. A000292, A040977, A133112.

%K easy,nonn,easy

%O 1,5

%A _Peter Bala_, Sep 13 2007

%E More terms from _Robert G. Wilson v_, Sep 17 2007