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%I #19 Aug 20 2021 06:42:28
%S 1,2,6,22,98,516,3172,22436,180252,1624888,16258376,178877832,
%T 2146674136,27907332272,390705042288,5860585983856,93769421948432,
%U 1594080384922656,28693447925921632
%N E.g.f.: exp(x+x^2/2)/(1-x).
%C A jeweler creates collections of necklaces using exactly n different colored beads ( to make the entire collection) then chooses some (or none or all) of the necklaces to sell. [From Geoffrey Critzer, Apr 20 2009]
%H Vincenzo Librandi, <a href="/A130907/b130907.txt">Table of n, a(n) for n = 0..200</a>
%F a(n)=n!+(sum(m=0..n, sum(k=1..m, (binomial(k,m-k)*2^(k-m))/k!)))*n!. [From Vladimir Kruchinin, Jul 02 2011]
%F D-finite with recurrence a(n) = (n+1)*a(n-1) - (n-2)*(n-1)*a(n-3) . - _Vaclav Kotesovec_, Oct 20 2012
%F a(n) ~ n!*exp(3/2) . - _Vaclav Kotesovec_, Oct 20 2012
%t CoefficientList[Series[Exp[x + x^2/2 - Log[1 - x]], {x, 0, 20}], x]* Table[n!, {n, 0, 20}] [From Geoffrey Critzer, Apr 20 2009]
%o (PARI) x='x+O('x^66); /* that many terms */
%o egf=exp(x+x^2/2)/(1-x);
%o Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Jul 11 2011 */
%Y Cf. A130905.
%K nonn
%O 0,2
%A _Karol A. Penson_, Jun 08 2007
%E I deleted the initial 1. - _Geoffrey Critzer_, Apr 19 2009
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