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a(n) = Sum_{k=0..n} Stirling2(n,k)*(1+(-1)^k)*2^k/2.
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%I #31 Oct 13 2022 07:27:36

%S 1,0,4,12,44,220,1228,7196,45004,303900,2201676,16920860,136966860,

%T 1163989788,10364408140,96463232284,935872773068,9440653262620,

%U 98809201693260,1071131795708188,12007932126074060

%N a(n) = Sum_{k=0..n} Stirling2(n,k)*(1+(-1)^k)*2^k/2.

%C Stirling transform of A199572 (aerated powers of 4).

%H Vincenzo Librandi, <a href="/A065143/b065143.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F Representation as a sum of an infinite series: a(n) = exp(2)*Sum_{k = 0..infinity} ((2*k)^n*2^(2*k)/(2*k)!) - sinh(2)*sum_{k = 0..infinity}(k^n*2^k/k!), for n >= 0.

%F E.g.f.: cosh(2*exp(x)-2). - _Vladeta Jovovic_, Sep 14 2003

%F From _Vaclav Kotesovec_, Aug 06 2014: (Start)

%F a(n) ~ n^n * cosh(2*exp(r)-2) / (r^n * (exp(n) * sqrt(4*exp(2*r)*r^2/n + 1-n+r))), where r is the root of the equation -2*exp(r)*r*tanh(2-2*exp(r)) = n.

%F (a(n)/n!)^(1/n) ~ exp(1/LambertW(n/2)) / LambertW(n/2).

%F (End)

%F a(n) = (Bell_n(2) + Bell_n(-2))/2, where Bell_n(x) is n-th Bell polynomial. - _Vladimir Reshetnikov_, Nov 01 2015

%F a(n) = 1; a(n) = 4 * Sum_{k=0..n-1} binomial(n-1, k) * A357598(k). - _Seiichi Manyama_, Oct 12 2022

%t Table[Sum[StirlingS2[n,k]*(1+(-1)^k)*2^k/2,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 06 2014 *)

%t Table[(BellB[n, 2] + BellB[n, -2])/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)

%o (PARI) a(n) = sum(k=0, n, stirling(n,k,2)*(1+(-1)^k)*2^k/2); \\ _Michel Marcus_, Nov 02 2015

%o (PARI) x='x+O('x^50); Vec(serlaplace(cosh(2*exp(x)-2))) \\ _G. C. Greubel_, Nov 16 2017

%Y Column k=4 of A357681.

%Y Cf. A199572, A264036, A357598.

%K nonn

%O 0,3

%A _Karol A. Penson_, Oct 17 2001