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%I #42 Feb 10 2022 13:38:16
%S 0,1,6,39,292,2505,24306,263431,3154824,41368977,589410910,9064804551,
%T 149641946796,2638693215769,49490245341642,983607047803815,
%U 20646947498718736,456392479671188001,10595402429677269174,257723100178182605287,6553958557721713088820
%N LAH transform of squares.
%C If the e.g.f. of b(n) is E(x) and a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*b(k), then the e.g.f. of a(n) is E(x/(1-x))/(1-x). - _Vladeta Jovovic_, Apr 16 2005
%C a(n) is the total number of elements in all partial permutations (injective partial functions) of {1,2,...,n} that are in a cycle. A fixed point is considered to be in a cycle. a(n) = Sum_{k=0..n} A206703(n,k)*k. - _Geoffrey Critzer_, Feb 11 2012
%C a(n) is the total number of elements in all partial permutations (injective partial functions) of {1,2,...,n} that are undefined, i.e., they do not have an image.- _Geoffrey Critzer_, Feb 09 2022
%C a(n) is the total length of all increasing subsequences over all n-permutations. Cf. A002720. - _Geoffrey Critzer_, Feb 09 2022
%H Alois P. Heinz, <a href="/A103194/b103194.txt">Table of n, a(n) for n = 0..200</a>
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 132.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = Sum_{k=0..n} (n!/k!)*binomial(n-1, k-1)*k^2.
%F E.g.f.: x/(1-x)^2*exp(x/(1-x)).
%F Recurrence: (n-1)*a(n) - n*(2*n-1)*a(n-1) + n*(n-1)^2*a(n-2) = 0.
%F a(n) = n*A000262(n). - _Vladeta Jovovic_, Mar 20 2005
%F a(n) ~ n! * exp(-1/2 + 2*sqrt(n))*n^(1/4)/(2*sqrt(Pi)). - _Vaclav Kotesovec_, Aug 13 2013
%F a(n) = n!*hypergeom([2, 1-n], [1, 1], -1). - _Peter Luschny_, Mar 30 2015
%p with(combstruct): SetSeqSetL := [T, {T=Set(S), S=Sequence(U, card >= 1), U=Set(Z, card=1)}, labeled]: seq(k*count(SetSeqSetL, size=k), k=0..18); # _Zerinvary Lajos_, Jun 06 2007
%p a := n -> n!*hypergeom([2, 1-n], [1, 1], -1):
%p seq(simplify(a(n)),n=0..20); # _Peter Luschny_, Mar 30 2015
%t nn = 20; a = 1/(1 - x); ay = 1/(1 - y x); D[Range[0, nn]! CoefficientList[ Series[Exp[a x] ay, {x, 0, nn}], x], y] /. y -> 1 (* _Geoffrey Critzer_, Feb 11 2012 *)
%Y Cf. A000262, A000290, A001477, A206703.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Mar 18 2005
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