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

Author

Peter Bala, Sep 13 2007

Keywords

• easy
• nonn

Comments

Compare with A000292 and A040977 for the corresponding sums for the Vandermonde matrices of orders 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
• Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Crossrefs

Cf. A000292, A040977, A107396, A133112.

Programs

Magma

A133111:= func< n | Binomial(n+2,5)*Binomial(n+4,5)/21 >;
[A133111(n): n in [0..30]]; // G. C. Greubel, Feb 09 2025

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 *)

SageMath

def A133111(n): return binomial(n+2,5)*binomial(n+4,5)//21
print([A133111(n) for n in range(31)]) # G. C. Greubel, Feb 09 2025

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.

a(n) = (1/21)*A107396(n-3). - G. C. Greubel, Feb 09 2025

Extensions

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