A052859 Expansion of e.g.f.: exp(exp(2*x) - 2*exp(x) + 1).

1, 0, 2, 6, 26, 150, 962, 6846, 54266, 471750, 4439762, 44911086, 485570186, 5581383990, 67890295202, 870493380126, 11726471352986, 165475293394470, 2439632685738482, 37491028556508366, 599285435979866666, 9945441791592272790, 171062503783616702402

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Previous name was: A simple grammar.

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 827
• Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022 (set m=1, b=2, r=-2, d=1, s=1).
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A060311, A067994.

Programs

Maple

spec := [S,{B=Prod(C,C),C=Set(Z,1 <= card),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *2*binomial(n-1, j-1)*Stirling2(j, 2), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2019

Mathematica

nn=20; a=Exp[x]-1; Range[0,nn]! CoefficientList[Series[Exp[a^2], {x,0,nn}], x]  (* Geoffrey Critzer, Jan 20 2012 *)
Table[Sum[BellY[n, k, 2^Range[n] - 2], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
Table[Sum[(2*k)!*StirlingS2[n, 2*k]/k!, {k, 0, n/2}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 04 2022 *)

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

PARI

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/k!); \\ Seiichi Manyama, May 07 2022

Formula

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

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

From Seiichi Manyama, May 07 2022: (Start)

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

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

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

Extensions

New name using e.g.f. from Vaclav Kotesovec, Oct 04 2022