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| A133111
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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.
(history;
published version)
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#20 by Alois P. Heinz at Sat May 11 19:11:44 EDT 2024
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#19 by Jon E. Schoenfield at Sat May 11 19:07:25 EDT 2024
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#18 by Jon E. Schoenfield at Sat May 11 19:07:22 EDT 2024
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| NAME
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a(n) = ) = (1/(1!*2!*3!*4!)*sum {!))*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.
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| COMMENTS
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Compare with A000292 and A040977 for the corresponding sums for the Vandermonde matrices of orderorders 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
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| FORMULA
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a(n) = ) = (1/288*sum {)*Sum_{1 <= i,j,k,l <= n} |(i-j)(i-k)(j-k)(i-l)(j-l)(k-l)|.
a(n) = sum {Sum_{i + j + k + l = n} i*j*k^2*l^3.
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| STATUS
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approved
editing
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#17 by Michael De Vlieger at Fri Mar 08 09:01:36 EST 2024
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#16 by Joerg Arndt at Fri Mar 08 02:51:44 EST 2024
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#15 by Joerg Arndt at Fri Mar 08 02:51:41 EST 2024
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#14 by Joerg Arndt at Fri Mar 08 02:51:39 EST 2024
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| MATHEMATICA
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fa[n_] := n^2 (n^2 - 1)^2 (n^2 - 4) (n^2 - 9)/302400; Array[fa, 30] (* or *) (* _] (* _Robert G. Wilson v_, Sep 17 2007 *)
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| STATUS
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proposed
editing
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#13 by Joerg Arndt at Fri Mar 08 02:51:11 EST 2024
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#12 by Joerg Arndt at Fri Mar 08 02:51:07 EST 2024
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| FORMULA
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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 + 5x + 5x^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.)|.
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.
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| KEYWORD
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easy,nonn,changed,easy
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| STATUS
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proposed
editing
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#11 by Michel Marcus at Fri Mar 08 02:43:18 EST 2024
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