A052761 a(n) = 3!*n*S(n-1,3), where S denotes the Stirling numbers of second kind.
0, 0, 0, 0, 24, 180, 900, 3780, 14448, 52164, 181500, 615780, 2052072, 6749028, 21976500, 71007300, 228009696, 728451972, 2317445100, 7346047140, 23213772120, 73156412196, 229989358500, 721474964100, 2258832312144, 7059480120900, 22026886599900
Offset: 0
Links
- Matthew House, Table of n, a(n) for n = 0..2079
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 718
- Index entries for linear recurrences with constant coefficients, signature (12,-58,144,-193,132,-36).
Programs
-
Maple
spec := [S,{B=Set(Z,1 <= card),S=Prod(B,B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20); -
Mathematica
Join[{0},Table[3!*n*StirlingS2[n-1,3],{n,30}]] (* Harvey P. Dale, Feb 07 2015 *) -
PARI
a(n)={if(n>=1, 3!*n*stirling(n-1, 3, 2), 0)} \\ Andrew Howroyd, Aug 08 2020
Formula
E.g.f.: exp(x)^3*x - 3*exp(x)^2*x + 3*x*exp(x) - x.
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-36*n^2 - 66*n - 6*n^3 - 36)*a(n) + (11*n^3 + 55*n^2 + 66*n)*a(n+1) + (-6*n^3 - 24*n^2 - 18*n)*a(n+2) + (n^3 + 3*n^2 + 2*n)*a(n+3)}
For n>=2, a(n) = n*(3^(n-1) - 3*2^(n-1) + 3). - Vaclav Kotesovec, Nov 27 2012
O.g.f.: 12*x^4*(2 - 9*x + 11*x^2 - 3*x^3)/((1 - 3*x)^2*(1 - 2*x)^2*(1 - x)^2). - Matthew House, Feb 16 2017 [Corrected by Georg Fischer, May 19 2019]
From Andrew Howroyd, Aug 08 2020: (Start)
a(n) = n*A001117(n-1) for n > 1.
E.g.f.: x*(exp(x) - 1)^3. (End)
Extensions
Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001
More terms from Harvey P. Dale, Feb 07 2015