This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A200978 #27 Jan 03 2021 15:55:58
%S A200978 6,72,720,7200,75600,846720,10160640,130636800,1796256000,26345088000,
%T A200978 410983372800,6799906713600,118998367488000,2196892938240000,
%U A200978 42682491371520000,870722823979008000,18611700362551296000,416026243398205440000
%N A200978 Number of ways to arrange n books on 3 consecutive shelves leaving none of the shelves empty.
%C A200978 To derive a(n), we note that there are n! ways to arrange n books in a row and there are binomial(n-1,2) ways to place the n arranged books on 3 consecutive shelves (since binomial(n-1,2) is the number of compositions of n with 3 summands). Hence a(n) = n!*binomial(n-1,2) for n >= 3.
%C A200978 The number of ways to arrange n books on two nonempty bookshelves is given by A062119(n).
%F A200978 a(n) = n!*binomial(n-1,2) = n!*(n-1)*(n-2)/2, n >= 3.
%F A200978 a(n) = A156992(n,3).
%F A200978 E.g.f.: x^3/(1-x)^3.
%F A200978 a(n) = A001754(n)*3!. - _Geoffrey Critzer_, Sep 02 2013
%e A200978 a(4)=72 since there are 72 ways to arrange books b1, b2, b3, and b4 on 3 consecutive shelves s1, s2, and s3. Note that there are 24 arrangements with two books on shelf s_i (i=1,2,3) and one book on each of the other two shelves. (For instance, there are 12 ways to select and permute the two books for s1 and 2 ways to select the single books for s2 and s3.) Hence there are 3(24)=71 book arrangements.
%p A200978 seq(n!*C(n-1,2),n=3..20);
%t A200978 nn=20;Drop[Range[0,nn]!CoefficientList[Series[(x/(1-x))^3,{x,0,nn}],x],3] (* _Geoffrey Critzer_, Sep 02 2013 *)
%Y A200978 Cf. A156992.
%K A200978 nonn,easy
%O A200978 3,1
%A A200978 _Dennis P. Walsh_, Nov 26 2011