A272603 - OEIS

%I #18 Jul 22 2018 11:35:09

%S 1,1,2,4,10,26,196,1072,7484,42940,261496,1477136,15219832,134828344,

%T 1488515120,13692017536,130252442896,1123580329232,14639510308384,

%U 173489066401600,2528654220104096,31472160333513376,402634734214583872,4645625988351336704,25925035549644280991680

%N Number of permutations of [n] whose cycle lengths are factorials.

%H Alois P. Heinz, <a href="/A272603/b272603.txt">Table of n, a(n) for n = 0..455</a>

%F E.g.f.: exp( sum(n>=1, x^(n!) / n! ) ).

%p a:= proc(n) option remember; local r, f, i;

%p       if n=0 then 1 else r, f, i:= $0..2;

%p         while f<=n do r:= r +a(n-f)*(f-1)!*

%p           binomial(n-1, f-1); f, i:= f*i, i+1

%p         od; r

%p       fi

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 04 2016

%t nmax = 4; egf = Exp[Sum[x^n!/n!, {n, 1, nmax}]] + O[x]^(nmax! + 1); CoefficientList[egf, x]*Range[0, nmax!]! (* _Jean-François Alcover_, Feb 19 2017 *)

%o (PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(n=1,10,x^(n!)/n!))))

%Y Cf. A000142, A273001 (cycle lengths are Fibonacci numbers), A272602 (e.g.f.: exp( sum(n>=1, x^(n!) / n ) ) ), A273996, A317132.

%K nonn

%O 0,3

%A _Joerg Arndt_, May 29 2016