OFFSET
1,2
REFERENCES
R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..440
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics, Vol. 10, No. 7 (2022), 1161.
FORMULA
a(n) = n*(n + 1)*(n + 3)!/48.
Essentially Stirling numbers of second kind - see A028246.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n-3) = (-1)^n*f(n,4,-3), (n>=4). - Milan Janjic, Mar 01 2009
E.g.f.: t*(3*t + 2)/(2*(t - 1)^6). - Ran Pan, Jul 10 2016
a(n) ~ sqrt(Pi/2)*exp(-n)*n^(n+1/2)*(n^5/24 + 85*n^4/288 + 5065*n^3/6912 + 955841*n^2/1244160 + 3710929*n/11943936). - Ilya Gutkovskiy, Jul 10 2016
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 16*(e + gamma - Ei(1)) - 64/3, where e = A001113, gamma = A001620, and Ei(1) = A091725.
Sum_{n>=1} (-1)^(n+1)/a(n) = 32*(gamma - Ei(-1)) - 16/e - 56/3, where Ei(-1) = -A099285. (End)
a(n) = (n-1)! * Stirling2(n+3, n). - G. C. Greubel, Nov 23 2022
EXAMPLE
G.f. = x + 15*x^2 + 180*x^3 + 2100*x^4 + 25200*x^5 + 317520*x^6 + ...
MAPLE
a:=n->sum((n-j)*n!/4!, j=3..n): seq(a(n), n=4..17); # Zerinvary Lajos, Apr 29 2007
MATHEMATICA
Table[(n(n+1)(n+3)!)/48, {n, 20}] (* Harvey P. Dale, Mar 14 2012 *)
a[ n_] := If[ n < 0, 0, n (n + 1) (n + 3)! / 48]; (* Michael Somos, May 27 2014 *)
PROG
(Sage) [factorial(m+1)*binomial(m-1, 2)/24 for m in range(3, 19)] # Zerinvary Lajos, Jul 05 2008
(Sage) [binomial(n, 4)*factorial (n-2)/2 for n in range(4, 18)] # Zerinvary Lajos, Jul 07 2009
(Magma) [Factorial(n-1)*StirlingSecond(n+3, n): n in [1..35]]; // G. C. Greubel, Nov 23 2022
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, Mar 14 2012
STATUS
approved