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 A038720 #47 May 12 2025 00:25:31
%S A038720 2,5,18,84,480,3240,25200,221760,2177280,23587200,279417600,
%T A038720 3592512000,49816166400,741015475200,11769069312000,198766503936000,
%U A038720 3556874280960000,67224923910144000,1338096104497152000,27978373094031360000,613091306060513280000
%N A038720 a(n) = (n+3)*n!/2.
%C A038720 Next-to-last diagonal of A038719.
%C A038720 a(n-1) is the sum of the n-th entries in all cycles of all permutations of [n]. a(2) = 5 because the sum of the third entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 3+2+0+0+0+0 = 5. - _Alois P. Heinz_, May 03 2017
%H A038720 Reinhard Zumkeller, <a href="/A038720/b038720.txt">Table of n, a(n) for n = 1..1000</a>
%H A038720 Rajesh Kumar Mohapatra and Tzung-Pei Hong, <a href="https://doi.org/10.3390/math10071161">On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences</a>, Mathematics (2022) Vol. 10, No. 7, 1161.
%H A038720 R. B. Nelsen and H. Schmidt, Jr., <a href="http://www.jstor.org/stable/2690450">Chains in power sets</a>, Math. Mag., 64 (1991), 23-31.
%H A038720 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>.
%F A038720 a(n) = A052572(n)/2.
%F A038720 a(n) = A214178(n+3,n). - _Reinhard Zumkeller_, Jul 08 2012
%F A038720 G.f.: Sum_{n>=1} ( (n+1)*x/(1 + (n+1)*x) )^n. - _Paul D. Hanna_, Jan 02 2013
%F A038720 E.g.f.: 1/(1-x) + 1/(2*(x-1)^2) - 3/2. - _Alois P. Heinz_, May 04 2017
%F A038720 From _Amiram Eldar_, Dec 11 2022: (Start)
%F A038720 Sum_{n>=1} 1/a(n) = 2*e - 14/3.
%F A038720 Sum_{n>=1} (-1)^(n+1)/a(n) = 10/e - 10/3. (End)
%t A038720 Array[(# + 3) #!/2 &, 21] (* _Michael De Vlieger_, Apr 28 2022 *)
%o A038720 (Haskell)
%o A038720 import Data.List (transpose)
%o A038720 a038720 n = a038720_list !! (n-1)
%o A038720 a038720_list = (transpose $ map reverse a038719_tabl) !! 1
%o A038720 -- _Reinhard Zumkeller_, Jul 08 2012
%o A038720 (Magma)
%o A038720 A038720:= func< n | (n+3)*Factorial(n)/2 >; // _G. C. Greubel_, May 11 2025
%o A038720 (SageMath)
%o A038720 def A038720(n): return (n+3)*factorial(n)//2 # _G. C. Greubel_, May 11 2025
%Y A038720 Cf. A038719, A038721, A052572, A214178.
%Y A038720 Main diagonal of A285793.
%K A038720 nonn,easy
%O A038720 1,1
%A A038720 _N. J. A. Sloane_, May 02 2000
%E A038720 Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 09 2000.