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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.

%I A133111 #27 May 22 2025 03:01:46
%S A133111 0,0,0,1,16,126,672,2772,9504,28314,75504,184041,416416,884884,
%T A133111 1782144,3426384,6325632,11267532,19442016,32605881,53300016,85131970,
%U A133111 133138720,204246900,307850400,456528150,666928080,960846705,1366537536,1920285576,2668289536
%N 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.
%C A133111 Compare with A000292 and A040977 for the corresponding sums for the Vandermonde matrices of orders 2 and 3 respectively.
%C A133111 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 A133111 T. D. Noe, <a href="/A133111/b133111.txt">Table of n, a(n) for n = 1..1000</a>
%H A133111 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).
%F A133111 a(n) = (1/288)*Sum_{1 <= i,j,k,l <= n} |(i-j)(i-k)(j-k)(i-l)(j-l)(k-l)|.
%F A133111 G.f.: x^4*(1 + 5*x + 5*x^2 + x^3)/(1 - x)^11 .
%F A133111 a(n) = n^2*(n^2 - 1)^2*(n^2 - 4)*(n^2 - 9)/302400.
%F A133111 a(n) = Sum_{i + j + k + l = n} i*j*k^2*l^3.
%F A133111 a(n) = (1/21)*A107396(n-3). - _G. C. Greubel_, Feb 09 2025
%t A133111 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 A133111 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 *)
%o A133111 (Magma)
%o A133111 A133111:= func< n | Binomial(n+2,5)*Binomial(n+4,5)/21 >;
%o A133111 [A133111(n): n in [0..30]]; // _G. C. Greubel_, Feb 09 2025
%o A133111 (SageMath)
%o A133111 def A133111(n): return binomial(n+2,5)*binomial(n+4,5)//21
%o A133111 print([A133111(n) for n in range(31)]) # _G. C. Greubel_, Feb 09 2025
%Y A133111 Cf. A000292, A040977, A107396, A133112.
%K A133111 easy,nonn
%O A133111 1,5
%A A133111 _Peter Bala_, Sep 13 2007
%E A133111 More terms from _Robert G. Wilson v_, Sep 17 2007