%I #33 Jun 26 2022 02:58:59

%S 1,0,2,2,14,42,222,1066,6078,36490,238046,1653610,12214270,95361866,

%T 784071966,6764984362,61066919230,575200190986,5640081557598,

%U 57450510336234,606773139773054,6633515763375306,74950634205257630,873995513192234410,10504736507220958142,129983468625156713354

%N a(n) = Sum_{k=0..n} binomial(n,k)*w(k)*w(n-k) where w() = A000296().

%D D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 771, Problem 37).

%H Seiichi Manyama, <a href="/A194689/b194689.txt">Table of n, a(n) for n = 0..560</a>

%F G.f.: 1/Q(0) where Q(k) = 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 07 2013

%F G.f.: 1/Q(0), where Q(k)= 1 - x*k - 2*x^2*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 06 2013

%F E.g.f.: exp(2*(exp(x) - 1 - x)). - _Ilya Gutkovskiy_, Apr 07 2018

%F a(0) = 1; a(n) = 2 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-1-k). - _Seiichi Manyama_, Nov 20 2020

%F a(n) ~ 4 * n^(n-2) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-2)). - _Vaclav Kotesovec_, Jun 26 2022

%t Table[Sum[(-1)^(n-k) * Binomial[n,k] * BellB[k,2] * 2^(n-k), {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Jun 25 2022 *)

%o (PARI)

%o N=66; x='x+O('x^N);

%o Q(k) = if (k>N, 1, 1 + x + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ) );

%o gf=1/Q(0); Vec(Ser(gf))

%o /* _Joerg Arndt_, Mar 07 2013 */

%o (PARI) my(N=66, x='x+O('x^N)); Vec(serlaplace(exp(2*(exp(x)-1-x)))) \\ _Seiichi Manyama_, Nov 20 2020

%Y Cf. A000296, A001861, A194689, A339014, A339017, A339027.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Sep 01 2011