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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.
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4
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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
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
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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.
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MATHEMATICA
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f[n_] := n^2 (n^2 - 1)^2 (n^2 - 4) (n^2 - 9)/302400; Array[f, 30] (* or *) - 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
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CROSSREFS
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Cf. A000292, A040977, A133112.
Sequence in context: A213748 A007787 A067470 * A268998 A253319 A291460
Adjacent sequences: A133108 A133109 A133110 * A133112 A133113 A133114
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KEYWORD
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easy,nonn
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AUTHOR
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Peter Bala, Sep 13 2007
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EXTENSIONS
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More terms from Robert G. Wilson v, Sep 17 2007
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STATUS
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approved
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