OFFSET
1,2
COMMENTS
From Dixon: The sequence is asymptotic to (n!)^2; when divided by n!^2, it has a high-order asymptotic contact with the probability that two randomly chosen permutations generate the symmetric group. Also: a(n)=(n-1)!*A003319(n+1), where A003319 is the number of connected [or indecomposable] permutations. The coefficients in the asymptotic expansion of a(n)/(n!)^2 are A113869 and in absolute value, they constitute A084357 (number of sets of sets of lists).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..253
John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 139.
FORMULA
Exponential generating function is: log(1+Sum_{n>=1}n!*z^n).
a(n) = (n!)^2 - (n-1)! * Sum_{k=1..n-1} a(k) * (n-k)! / (k-1)!. - Ilya Gutkovskiy, Jul 10 2020
EXAMPLE
a(2)=3 because there are 2!*2!=4 pairs of permutations, of which only [(1,1),(1,1)] does not generate a transitive group.
MAPLE
series(log(add(n!*z^n, n=0..Order+2)), z=0):seq(coeff(%, z, j)*j!, j=0..Order);
MATHEMATICA
max = 15; Drop[ CoefficientList[ Series[ Log[1 + Sum[n!*z^n, {n, 1, max}]], {z, 0, max}], z]* Range[0, max]!, 1](* Jean-François Alcover, Oct 05 2011 *)
PROG
(PARI) N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, k!*x^k)))) \\ Seiichi Manyama, Mar 01 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Flajolet, Oct 25 2006
EXTENSIONS
More terms from Seiichi Manyama, Mar 01 2019
STATUS
approved