A121723 a(n) = A098916(n+2) + (1-n) * A067318(n).

0, 3, 22, 150, 1096, 8820, 78408, 767088, 8212608, 95657760, 1205438400, 16350871680, 237633108480, 3685053415680, 60748282022400, 1061014235904000, 19574489449267200, 380408796994867200, 7768172642717491200

Offset: 1

Author

Joel Duet, Aug 17 2006

Keywords

• nonn

Comments

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.

Examples

			a(3) = 22 because we can write 22 = A098916(3) + (1-3) * A067318(3) = 36 - 2*7.
		

Crossrefs

Cf. A067318, A098916.

Programs

Mathematica

f[n_] := n! (n - 1) HarmonicNumber[n]; Array[f, 19] (* Robert G. Wilson v, Sep 07 2011 *)

Formula

a(n) = n!*(n-1)*Sum_{i=1..n} (1/i).

Extensions

Definition corrected by Gary Detlefs, Sep 07 2011