A088311 - OEIS

%I #34 Dec 15 2022 05:15:28

%S 1,1,2,12,48,360,2880,25200,241920,2903040,36288000,479001600,

%T 7185024000,112086374400,1917922406400,35307207936000,669529276416000,

%U 13516122267648000,294509190463488000,6568835422076928000,155705728523304960000,3882911605049917440000

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

%C 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

%H Vincenzo Librandi, <a href="/A088311/b088311.txt">Table of n, a(n) for n = 0..200</a>

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

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

%p b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(

%p      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

%p     end:

%p a:= n-> n!*b(n):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 15 2018

%t 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 *)

%t 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 *)

%o (PARI) my(x='x+O('x^66)); Vec(serlaplace(eta(x^2)/eta(x))) \\ _Joerg Arndt_, Aug 06 2013

%o (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

%o (SageMath)

%o # uses[EulerTransform from A166861]

%o a = BinaryRecurrenceSequence(0, 1) #  Peter Luschny's code of A000009 and A166861

%o b = EulerTransform(a)

%o [factorial(n)*b(n) for n in range(41)] # _G. C. Greubel_, Dec 14 2022

%Y Cf. A000009, A007837, A007838.

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

%K nonn

%O 0,3

%A _Vladeta Jovovic_, Nov 05 2003

%E Prepended a(0) = 1, _Joerg Arndt_, Aug 06 2013