A114329 - OEIS

%I #33 Feb 21 2022 15:14:51

%S 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,

%T 30,0,1,17640,13440,5040,1260,210,42,0,1,183120,141120,53760,13440,

%U 2520,336,56,0,1,2116800,1648080,635040,161280,30240,4536,504,72,0,1

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

%C 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)

%H Nathaniel Johnston, <a href="/A114329/b114329.txt">Table of n, a(n) for n = 0..5150</a>

%F 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).

%e Triangle begins:

%e     1;

%e     0,   1;

%e     2,   0,  1;

%e     6,   6,  0,  1;

%e    36,  24, 12,  0, 1;

%e   240, 180, 60, 20, 0, 1;

%e   ...

%p 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

%p # second Maple program:

%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*

%p      `if`(j=1, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):

%p seq(T(n), n=0..10);  # _Alois P. Heinz_, Feb 19 2022

%t 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 *)

%Y Cf. A000262, A006152, A052845, A052852.

%K easy,nonn,tabl

%O 0,4

%A _Vladeta Jovovic_, Feb 06 2006