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

0,4

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)

Links

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

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).

Example

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

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 *)

Crossrefs

Cf. A000262, A006152, A052845, A052852.

Sequence in context: A138106 A131689 A278075 * A361956 A241011 A220905

Adjacent sequences: A114326 A114327 A114328 * A114330 A114331 A114332

Keyword

easy,nonn,tabl

Author

Vladeta Jovovic, Feb 06 2006

Status

approved