OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
FORMULA
a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A048144(k).
G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^n.
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.4670932578797312973586879293426... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 2^(log(2)/2-2) / (log(2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
G.f.: Sum_{n>=0} (1-x)^n * (1 - (1-x)^n)^n. - Paul D. Hanna, Mar 26 2018
EXAMPLE
From Gus Wiseman, Nov 14 2018: (Start)
The a(3) = 15 matrices:
[3]
.
[2 0] [1 1] [1 1] [1 0] [1 0] [0 2] [0 1] [0 1]
[0 1] [1 0] [0 1] [1 1] [0 2] [1 0] [2 0] [1 1]
.
[1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
[0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
[0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
(End)
MATHEMATICA
Table[1/n!*Sum[(-1)^(n-k)*StirlingS1[n, k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2014 *)
multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n], 2], n], Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]], {n, 5}] (* Gus Wiseman, Nov 14 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Aug 18 2006
STATUS
approved