A086325 Let u(1)=0, u(2)=1, u(k)=u(k-1)+u(k-2)/(k-2); then a(n)=n!*u(n).

0, 2, 6, 36, 220, 1590, 12978, 118664, 1201464, 13349610, 161530270, 2114578092, 29780308116, 448995414686, 7215997736010, 123153028027920, 2224451568754288, 42395429898611154, 850263899633257014, 17900292623858042420, 394701452356069835340, 9096928711444657157382, 218739785834282892557026

Offset: 1

Author

N. J. A. Sloane. This sequence appeared in the 1973 "Handbook", but was then omitted from the database. Resubmitted by Benoit Cloitre, Aug 30 2003. Entry revised by N. J. A. Sloane, Jun 11 2012

Keywords

• nonn

References

• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263, Table 7.5.1, row 3.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210, Table 3, Three-line Latin rectangles.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.

Crossrefs

Cf. A000246, A000274, A000166.

Programs

Maple

a:=n->n!*add((-1)^k/k!, k=0..n): seq(a(n)*n, n=1..19); # Zerinvary Lajos, Dec 18 2007
with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*fibonacci(2,n), n=1..19); # Zerinvary Lajos, Jun 11 2008
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*n, n=1..19); # Zerinvary Lajos, Jun 11 2008

Mathematica

Table[Subfactorial[n]*n, {n, 1, 19}] (* Zerinvary Lajos, Jul 09 2009 *)

PARI

a(n) = n*((n! + 1)\exp(1)); \\ Indranil Ghosh, Apr 13 2017

Formula

a(n) = ceiling(n*n!/e) - (1-(-1)^n)/2.

E.g.f.: x^2*exp(-x)/(1-x)^2. - Vladeta Jovovic, Nov 20 2003

a(n) = n*floor((n!+1)/e). [Gary Detlefs, Jul 13 2010]

a(n) = n * A000166(n). [Joerg Arndt, Jul 09 2012]

G.f.: x*f'(x), where f(x) = 1/(1 + x) + Sum_{k>=1} k^k*x^k/(1 + (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017