A114329 Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.

1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 36, 24, 12, 0, 1, 240, 180, 60, 20, 0, 1, 1920, 1440, 540, 120, 30, 0, 1, 17640, 13440, 5040, 1260, 210, 42, 0, 1, 183120, 141120, 53760, 13440, 2520, 336, 56, 0, 1, 2116800, 1648080, 635040, 161280, 30240, 4536, 504, 72, 0, 1

Offset: 0

Author

Vladeta Jovovic, Feb 06 2006

Keywords

• easy
• nonn
• tabl

Comments

The average number of size 1 lists goes to 1 as n->infinity. In other words, lim_{n->infinity} Sum_{k>=1} T(n,k)*k/A000262(n) = 1. - Geoffrey Critzer, Feb 20 2022 (after asymptotic limits by Vaclav Kotesovec given in A000262)

Examples

			Triangle begins:
    1;
    0,   1;
    2,   0,  1;
    6,   6,  0,  1;
   36,  24, 12,  0, 1;
  240, 180, 60, 20, 0, 1;
  ...
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..5150

Crossrefs

Cf. A000262, A006152, A052845, A052852.

Programs

Maple

t:=taylor(exp(x/(1-x)+(y-1)*x),x,11):for n from 0 to 10 do for k from 0 to n do printf("%d, ",coeff(n!*coeff(t,x,n),y,k)): od: printf("\n"): od: # Nathaniel Johnston, Apr 27 2011
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
     `if`(j=1, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 19 2022

Mathematica

nn = 10; Table[Take[(Range[0, nn]! CoefficientList[ Series[Exp[ x/(1 - x) - x + y x], {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}] // Grid (* Geoffrey Critzer, Feb 19 2022 *)

Formula

E.g.f.: exp(x/(1-x)+(y-1)*x). More generally, e.g.f. for number of partitions of n-set into lists with k lists of size m is exp(x/(1-x)+(y-1)*x^m).