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0, 3, 22, 150, 1096, 8820, 78408, 767088, 8212608, 95657760, 1205438400, 16350871680, 237633108480, 3685053415680, 60748282022400, 1061014235904000, 19574489449267200, 380408796994867200, 7768172642717491200
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OFFSET
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1,2
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COMMENTS
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This sequence arises when evaluating the generalized sub-volumes of the linearly weighted (n-1)-simplex in dimension n-1. For instance, in dimension 1 where n=2, the 1-simplex is the interval [H;J] of the real line (we suppose H < J). When H is weighted by the real h and J by j, the signed surface of the polygon {(H,0),(J,0),(J,j),(H,h)} of the Euclidean plane is S = (h+j)/2*(J-H).
Then we consider I the middle of [H;J]. It is linearly weighted by i = (h+j)/2. When we search for the weights w1(2) and w2(2) so that the 2 equations 2*Sh/(J-H) = h*w1(2) + j*w2(2) = (h+i)/2 and 2*Sj/(J-H) = h*w2(2) + j*w1(2) = (i+j)/2 are verified (which implies Sh + Sj = S also), we find that w1(2) = a(2)/A098916(2) = 3/4 and w2(2) = A067318(2)/A098916(2) = 1/4.
Even in higher dimensions (n > 2), there are only 2 weights: one for the considered sub-volume and the other for the other sub-volumes. For instance, in dimension 2 where n=3, the first weight w1(3) = 11/18 refers to the part of the triangle which is delimited by the 4 points: one top A, then the middle of [A;B], then the center of gravity, then the middle of [A;C]; and w2(3) = 7/36 refers to any of the 2 other parts of the triangle.
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LINKS
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FORMULA
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a(n) = n!*(n-1)*Sum_{i=1..n} (1/i).
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EXAMPLE
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a(3) = 22 because we can write 22 = A098916(3) + (1-3) * A067318(3) = 36 - 2*7.
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Joel Duet (joel.duet(AT)free.fr), Aug 17 2006
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EXTENSIONS
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STATUS
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approved
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