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

%I A088311 #34 Dec 15 2022 05:15:28
%S A088311 1,1,2,12,48,360,2880,25200,241920,2903040,36288000,479001600,
%T A088311 7185024000,112086374400,1917922406400,35307207936000,669529276416000,
%U A088311 13516122267648000,294509190463488000,6568835422076928000,155705728523304960000,3882911605049917440000
%N A088311 Number of sets of lists with distinct list sizes, cf. A000262.
%C A088311 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 A088311 Vincenzo Librandi, <a href="/A088311/b088311.txt">Table of n, a(n) for n = 0..200</a>
%F A088311 E.g.f: Product_{m>0} (1+x^m).
%F A088311 a(n) = n! * A000009(n).
%p A088311 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A088311      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A088311     end:
%p A088311 a:= n-> n!*b(n):
%p A088311 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 15 2018
%t A088311 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 A088311 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 A088311 (PARI) my(x='x+O('x^66)); Vec(serlaplace(eta(x^2)/eta(x))) \\ _Joerg Arndt_, Aug 06 2013
%o A088311 (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 A088311 (SageMath)
%o A088311 # uses[EulerTransform from A166861]
%o A088311 a = BinaryRecurrenceSequence(0, 1) #  Peter Luschny's code of A000009 and A166861
%o A088311 b = EulerTransform(a)
%o A088311 [factorial(n)*b(n) for n in range(41)] # _G. C. Greubel_, Dec 14 2022
%Y A088311 Cf. A000009, A007837, A007838.
%Y A088311 Other ordered permutation function pair relations are A000012, A000085, A000142, A001044, A053529.
%K A088311 nonn
%O A088311 0,3
%A A088311 _Vladeta Jovovic_, Nov 05 2003
%E A088311 Prepended a(0) = 1, _Joerg Arndt_, Aug 06 2013