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

0,3

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

Formula

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

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

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 *)

Prog

(PARI) my(x='x+O('x^66)); Vec(serlaplace(eta(x^2)/eta(x))) \\ Joerg Arndt, Aug 06 2013
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( (&*[1+x^j: j in [1..m+2]]) ))); // G. C. Greubel, Dec 14 2022
(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

Crossrefs

Cf. A000009, A007837, A007838.

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

Sequence in context: A232663 A052591 A029766 * A359118 A338522 A277183

Adjacent sequences: A088308 A088309 A088310 * A088312 A088313 A088314

Keyword

nonn

Author

Vladeta Jovovic, Nov 05 2003

Extensions

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

Status

approved