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 A038155 #33 Feb 16 2025 08:32:37
%S A038155 0,0,1,6,30,160,975,6846,54796,493200,4932045,54252550,651030666,
%T A038155 8463398736,118487582395,1777313736030,28437019776600,483429336202336,
%U A038155 8701728051642201,165332832981201990,3306656659624039990,69439789852104840000
%N A038155 a(n) = (n!/2) * Sum_{k=0..n-2} 1/k!.
%C A038155 For n>=2, a(n) gives the operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives the number of comparisons required to find the first interchangeable element in step L3 (see answer to exercise 5). - _Hugo Pfoertner_, Jan 27 2003
%C A038155 a(n) mod 5 = A011658(n+1). - _G. C. Greubel_, Apr 13 2016
%C A038155 a(450) has 1001 decimal digits. - _Michael De Vlieger_, Apr 13 2016
%C A038155 Also the number of (undirected) paths in the complete graph K_n. - _Eric W. Weisstein_, Jun 04 2017
%D A038155 D. E. Knuth: The Art of Computer Programming, Volume 4, Combinatorial Algorithms, Volume 4A, Enumeration and Backtracking. Pre-fascicle 2B, A draft of section 7.2.1.2: Generating all permutations. Available online; see link.
%H A038155 Michael De Vlieger, <a href="/A038155/b038155.txt">Table of n, a(n) for n = 0..449</a>
%H A038155 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/fasc2b.ps.gz">TAOCP Vol. 4, Pre-fascicle 2b (generating all permutations)</a>.
%H A038155 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/lpure.txt">FORTRAN implementation of Knuth's Algorithm L for lexicographic permutation generation</a>.
%H A038155 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>
%H A038155 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>
%F A038155 a(n) = 1/2*floor(n!*exp(1)-n-1), n>0. - _Vladeta Jovovic_, Aug 18 2002
%F A038155 E.g.f.: x^2/2*exp(x)/(1-x). - _Vladeta Jovovic_, Aug 25 2002
%F A038155 a(n) = Sum_{k=0..n-1} a(n-1) + k, a(0)=0. - _Ilya Gutkovskiy_, Apr 13 2016
%F A038155 a(n) = A038154(n)/2. - _Alois P. Heinz_, Jan 26 2017
%p A038155 A038155:=n->(n!/2)*add(1/k!, k=0..n-2): seq(A038155(n), n=0..30); # _Wesley Ivan Hurt_, Apr 16 2016
%t A038155 RecurrenceTable[{a[0] == 0, a[n] == Sum[a[n - 1] + k, {k, 0, n - 1}]}, a, {n, 21}] (* _Ilya Gutkovskiy_, Apr 13 2016 *)
%t A038155 Table[(n!/2) Sum[1/k!, {k, 0, n - 2}], {n, 0, 21}] (* _Michael De Vlieger_, Apr 13 2016 *)
%t A038155 Table[1/2 E (n - 1) n Gamma[n - 1, 1], {n, 0, 20}] (* _Eric W. Weisstein_, Jun 04 2017 *)
%t A038155 Table[If[n == 0, 0, Floor[n! E - n - 1]/2], {n, 0, 20}] (* _Eric W. Weisstein_, Jun 04 2017 *)
%Y A038155 Cf. A011658, A038154, A038156, A056542, A079752, A079884, A080047, A080048, A080049.
%K A038155 nonn,easy
%O A038155 0,4
%A A038155 _N. J. A. Sloane_