A088311 Number of sets of lists with distinct list sizes, cf. A000262.

1, 1, 2, 12, 48, 360, 2880, 25200, 241920, 2903040, 36288000, 479001600, 7185024000, 112086374400, 1917922406400, 35307207936000, 669529276416000, 13516122267648000, 294509190463488000, 6568835422076928000, 155705728523304960000, 3882911605049917440000

Offset: 0

Author

Vladeta Jovovic, Nov 05 2003

Keywords

• nonn

Comments

a(n) also enumerates ordered pairs of permutation functions on n elements where f(g(x)) = g(g(f(x))). - Chad Brewbaker, Mar 27 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000009, A007837, A007838.

Other ordered permutation function pair relations are A000012, A000085, A000142, A001044, A053529.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( (&*[1+x^j: j in [1..m+2]]) ))); // G. C. Greubel, Dec 14 2022

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> n!*b(n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

Mathematica

nn = 19; Drop[ Range[0, nn]! CoefficientList[ Series[ Product[1 + x^i, {i,nn}], {x,0,nn}], x], 0] (* Geoffrey Critzer, Aug 05 2013; adapted to new offset by Vincenzo Librandi, Mar 28 2014 *)
nmax = 20; CoefficientList[Series[Product[1/(1-x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 19 2015 *)

PARI

my(x='x+O('x^66)); Vec(serlaplace(eta(x^2)/eta(x))) \\ Joerg Arndt, Aug 06 2013

SageMath

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1) #  Peter Luschny's code of A000009 and A166861
b = EulerTransform(a)
[factorial(n)*b(n) for n in range(41)] # G. C. Greubel, Dec 14 2022

Formula

E.g.f: Product_{m>0} (1+x^m).

a(n) = n! * A000009(n).

Extensions

Prepended a(0) = 1, Joerg Arndt, Aug 06 2013