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URL: https://oeis.org/A114329

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A114329
Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.
2
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
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
# Alternative:
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
KEYWORD
easy,nonn,tabl
AUTHOR
Vladeta Jovovic, Feb 06 2006
STATUS
approved