A131631 Supersubfactorials: partial product of positive subfactorials (A000166).

1, 2, 18, 792, 209880, 389117520, 5771780174160, 770509566129663360, 1028600220910021528728960, 15104551945968674840127424147200, 2661646219535110627933754465838408595200

Offset: 2

Author

Jonathan Vos Post, Sep 01 2007

Keywords

• easy
• nonn

Comments

This is to subfactorials (A000166, rencontres numbers, or derangements) as superfactorials (A000178) are to factorials (A000142).

Examples

			a(2) = 1.
a(3) = 1 * 2 = 2.
a(4) = 1 * 2 * 9 = 18 = 2 * 3^2.
a(5) = 1 * 2 * 9 * 44 = 792 = 2^3 * 3^2 * 11.
a(6) = 1 * 2 * 9 * 44 * 265 = 209880 = 2^3 * 3^2 * 5 * 11 * 53.
a(7) = 1 * 2 * 9 * 44 * 265 * 1854 = 389117520 = 2^4 * 3^4 * 5 * 11 * 53 * 103.
		

Crossrefs

Cf. A000142, A000166, A000178.

Programs

Mathematica

Table[Product[k!*Sum[(-1)^j/j!, {j,0,k}], {k,2,n}], {n,2,15}] (* Vaclav Kotesovec, Jul 11 2015 *)

Formula

a(n) = Product_{k=2..n} A000166(k).

a(n) ~ c * n^(n^2/2 + n + 5/12) * (2*Pi)^((n+1)/2) / (A * exp(3*n^2/4 + 2*n - 13/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and c = 1.2517384488693662195086340541087053383189277225386098721341690164735... . - Vaclav Kotesovec, Jul 11 2015