A133111 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.

0, 0, 0, 1, 16, 126, 672, 2772, 9504, 28314, 75504, 184041, 416416, 884884, 1782144, 3426384, 6325632, 11267532, 19442016, 32605881, 53300016, 85131970, 133138720, 204246900, 307850400, 456528150, 666928080, 960846705, 1366537536, 1920285576, 2668289536

Offset

1,5

Comments

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

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Formula

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

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

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

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

Mathematica

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 *)
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 *)

Crossrefs

Cf. A000292, A040977, A133112.

Sequence in context: A213748 A007787 A067470 * A268998 A253319 A367070

Adjacent sequences: A133108 A133109 A133110 * A133112 A133113 A133114

Keyword

easy,nonn,easy

Author

Peter Bala, Sep 13 2007

Extensions

More terms from Robert G. Wilson v, Sep 17 2007

Status

approved