This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A065143 #32 Feb 16 2025 08:32:45
%S A065143 1,0,4,12,44,220,1228,7196,45004,303900,2201676,16920860,136966860,
%T A065143 1163989788,10364408140,96463232284,935872773068,9440653262620,
%U A065143 98809201693260,1071131795708188,12007932126074060
%N A065143 a(n) = Sum_{k=0..n} Stirling2(n,k)*(1+(-1)^k)*2^k/2.
%C A065143 Stirling transform of A199572 (aerated powers of 4).
%H A065143 Vincenzo Librandi, <a href="/A065143/b065143.txt">Table of n, a(n) for n = 0..200</a>
%H A065143 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%H A065143 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A065143 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 A065143 E.g.f.: cosh(2*exp(x)-2). - _Vladeta Jovovic_, Sep 14 2003
%F A065143 From _Vaclav Kotesovec_, Aug 06 2014: (Start)
%F A065143 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 A065143 (a(n)/n!)^(1/n) ~ exp(1/LambertW(n/2)) / LambertW(n/2).
%F A065143 (End)
%F A065143 a(n) = (Bell_n(2) + Bell_n(-2))/2, where Bell_n(x) is n-th Bell polynomial. - _Vladimir Reshetnikov_, Nov 01 2015
%F A065143 a(n) = 1; a(n) = 4 * Sum_{k=0..n-1} binomial(n-1, k) * A357598(k). - _Seiichi Manyama_, Oct 12 2022
%t A065143 Table[Sum[StirlingS2[n,k]*(1+(-1)^k)*2^k/2,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 06 2014 *)
%t A065143 Table[(BellB[n, 2] + BellB[n, -2])/2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)
%o A065143 (PARI) a(n) = sum(k=0, n, stirling(n,k,2)*(1+(-1)^k)*2^k/2); \\ _Michel Marcus_, Nov 02 2015
%o A065143 (PARI) x='x+O('x^50); Vec(serlaplace(cosh(2*exp(x)-2))) \\ _G. C. Greubel_, Nov 16 2017
%Y A065143 Column k=4 of A357681.
%Y A065143 Cf. A199572, A264036, A357598.
%K A065143 nonn
%O A065143 0,3
%A A065143 _Karol A. Penson_, Oct 17 2001