cp's OEIS Frontend

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

%I A121723 #30 Oct 12 2024 14:47:08
%S A121723 0,3,22,150,1096,8820,78408,767088,8212608,95657760,1205438400,
%T A121723 16350871680,237633108480,3685053415680,60748282022400,
%U A121723 1061014235904000,19574489449267200,380408796994867200,7768172642717491200
%N A121723 a(n) = A098916(n+2) + (1-n) * A067318(n).
%C A121723 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).
%C A121723 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.
%C A121723 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.
%F A121723 a(n) = n!*(n-1)*Sum_{i=1..n} (1/i).
%e A121723 a(3) = 22 because we can write 22 = A098916(3) + (1-3) * A067318(3) = 36 - 2*7.
%t A121723 f[n_] := n! (n - 1) HarmonicNumber[n]; Array[f, 19] (* _Robert G. Wilson v_, Sep 07 2011 *)
%Y A121723 Cf. A067318, A098916.
%K A121723 nonn
%O A121723 1,2
%A A121723 _Joel Duet_, Aug 17 2006
%E A121723 Definition corrected by _Gary Detlefs_, Sep 07 2011