A038720 a(n) = (n+3)*n!/2.

2, 5, 18, 84, 480, 3240, 25200, 221760, 2177280, 23587200, 279417600, 3592512000, 49816166400, 741015475200, 11769069312000, 198766503936000, 3556874280960000, 67224923910144000, 1338096104497152000, 27978373094031360000, 613091306060513280000

Offset: 1

Author

N. J. A. Sloane, May 02 2000

Keywords

• nonn
• easy

Comments

Next-to-last diagonal of A038719.

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
• R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
• Index entries for sequences related to posets.

Crossrefs

Cf. A038719, A038721, A052572, A214178.

Main diagonal of A285793.

Programs

Haskell

import Data.List (transpose)
a038720 n = a038720_list !! (n-1)
a038720_list = (transpose $ map reverse a038719_tabl) !! 1
-- Reinhard Zumkeller, Jul 08 2012

Magma

A038720:= func< n | (n+3)*Factorial(n)/2 >; // G. C. Greubel, May 11 2025

Mathematica

Array[(# + 3) #!/2 &, 21] (* Michael De Vlieger, Apr 28 2022 *)

SageMath

def A038720(n): return (n+3)*factorial(n)//2 # G. C. Greubel, May 11 2025

Formula

a(n) = A052572(n)/2.

a(n) = A214178(n+3,n). - Reinhard Zumkeller, Jul 08 2012

G.f.: Sum_{n>=1} ( (n+1)*x/(1 + (n+1)*x) )^n. - Paul D. Hanna, Jan 02 2013

E.g.f.: 1/(1-x) + 1/(2*(x-1)^2) - 3/2. - Alois P. Heinz, May 04 2017

From Amiram Eldar, Dec 11 2022: (Start)

Sum_{n>=1} 1/a(n) = 2*e - 14/3.

Sum_{n>=1} (-1)^(n+1)/a(n) = 10/e - 10/3. (End)

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 09 2000.