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Expansion of e.g.f.: exp(exp(2*x) - 2*exp(x) + 1).
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%I #48 Oct 04 2022 15:49:23

%S 1,0,2,6,26,150,962,6846,54266,471750,4439762,44911086,485570186,

%T 5581383990,67890295202,870493380126,11726471352986,165475293394470,

%U 2439632685738482,37491028556508366,599285435979866666,9945441791592272790,171062503783616702402

%N Expansion of e.g.f.: exp(exp(2*x) - 2*exp(x) + 1).

%C Previous name was: A simple grammar.

%C a(n) is the number of ways to select a nonempty proper subset from each block of the set partitions of {1,2,...,n}. - _Geoffrey Critzer_, Jan 20 2012

%H Vincenzo Librandi, <a href="/A052859/b052859.txt">Table of n, a(n) for n = 0..200</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=827">Encyclopedia of Combinatorial Structures 827</a>

%H Vaclav Kotesovec, <a href="https://arxiv.org/abs/2207.10568">Asymptotics for a certain group of exponential generating functions</a>, arXiv:2207.10568 [math.CO], Jul 13 2022 (set m=1, b=2, r=-2, d=1, s=1).

%H Vaclav Kotesovec, <a href="/A052859/a052859.jpg">Graph - the asymptotic ratio (10000 terms)</a>

%F E.g.f.: exp(exp(x)^2-2*exp(x)+1).

%F Stirling transform of unsigned Hermite numbers: Sum_{k=0..n} Stirling2(n, k)*|HermiteH(k, 0)|. - _Vladeta Jovovic_, Sep 12 2003

%F From _Seiichi Manyama_, May 07 2022: (Start)

%F G.f.: Sum_{k>=0} (2*k)! * x^(2*k)/(k! * Product_{j=1..2*k} (1 - j * x)).

%F a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/k!. (End)

%F a(n) ~ 2^n * exp(1/2 - n - 2*sqrt(n/LambertW(n)) + n/LambertW(n)) * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n). - _Vaclav Kotesovec_, Oct 04 2022

%p spec := [S,{B=Prod(C,C),C=Set(Z,1 <= card),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%p # second Maple program:

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

%p *2*binomial(n-1, j-1)*Stirling2(j, 2), j=1..n))

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 02 2019

%t nn=20; a=Exp[x]-1; Range[0,nn]! CoefficientList[Series[Exp[a^2], {x,0,nn}], x] (* _Geoffrey Critzer_, Jan 20 2012 *)

%t Table[Sum[BellY[n, k, 2^Range[n] - 2], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)

%t Table[Sum[(2*k)!*StirlingS2[n, 2*k]/k!, {k, 0, n/2}], {n, 0, 25}] (* _Vaclav Kotesovec_, Oct 04 2022 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*k)!*x^(2*k)/(k!*prod(j=1, 2*k, 1-j*x)))) \\ _Seiichi Manyama_, May 07 2022

%o (PARI) a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/k!); \\ _Seiichi Manyama_, May 07 2022

%Y Cf. A060311, A067994.

%K easy,nonn

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E New name using e.g.f. from _Vaclav Kotesovec_, Oct 04 2022