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%I #43 Jun 01 2021 11:59:41
%S 0,0,2,16,118,926,7956,75132,777456,8771184,107307360,1416252960,
%T 20068629120,304002322560,4903642679040,83928856838400,
%U 1519397749094400,29010025797580800,582647327132774400,12280347845905305600,271030782903552000000,6251213902855219200000
%N n! times the average number of comparisons required by an insertion sort of n (distinct) elements.
%H Alois P. Heinz, <a href="/A159324/b159324.txt">Table of n, a(n) for n = 0..448</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Insertion_sort">Insertion sort</a>
%H <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F a(n) = a(n-1)*(n) + n! *(n+1)/2 - (n-1)!.
%F a(n) = Sum_k A159323(n,k) = Sum_k A129178(n,k) * (n(n-1)/2 - k).
%F a(n) = n!/4 *(n^2+3*n-4*H(n)), where H(n) = Sum_{k=1..n} 1/k. - _Gary Detlefs_, Sep 02 2010
%F a(n) = A138772(n+1) - A000254(n). - _Gary Detlefs_, May 13 2012
%F a(n) = ((2*n^3-n^2-5*n+2)*a(n-1)-(n+2)*(n-1)^3*a(n-2))/((n-2)*(n+1)) for n>2. - _Alois P. Heinz_, Dec 16 2016
%F a(n) = 2 * A285231(n+1). - _Alois P. Heinz_, Apr 15 2017
%e For n=3, insertion sorting 123, 213, 213, 231, 312, 321 takes 3+3+3+2+3+2 = 4*3+2*2 = 16 comparisons.
%p a:= proc(n) option remember;
%p `if`(n<2, 0, a(n-1)*n + (n-1)! * (n-1)*(n+2)/2)
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 14 2012
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<3, [0$2, 2][n+1],
%p ((2*n^3-n^2-5*n+2)*a(n-1)-(n+2)*(n-1)^3*a(n-2))/((n-2)*(n+1)))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Dec 16 2016
%t a[n_] := n! ((n+1)(n+2)/4 - HarmonicNumber[n] - 1/2); Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 12 2017, after _Gary Detlefs_ *)
%Y Row sums of triangle A159323.
%Y Cf. A000254, A001008, A002805, A138772, A212395, A285231.
%K nonn
%O 0,3
%A Harmen Wassenaar (towr(AT)ai.rug.nl), Apr 10 2009
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