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 A126390 #41 Jun 27 2022 03:11:45
%S A126390 1,3,13,71,457,3355,27509,248127,2434129,25741939,291397789,
%T A126390 3510328695,44782460313,602513988107,8518757813637,126179029108463,
%U A126390 1952609274344353,31492811964616163,528249539951292461,9197240228562763687,165923214676585626729
%N A126390 a(n) = Sum_{i=0..n} 2^i*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n).
%H A126390 Vincenzo Librandi, <a href="/A126390/b126390.txt">Table of n, a(n) for n = 0..200</a>
%H A126390 Toufik Mansour and Mark Shattuck, <a href="https://www.emis.de/journals/INTEGERS/papers/l67/l67.Abstract.html">A recurrence related to the Bell numbers</a>, INTEGERS 11 (2011), #A67.
%F A126390 E.g.f.: exp(exp(2*x)-1+x). - _Vladeta Jovovic_, Aug 04 2007
%F A126390 a(n) = e^(-1)* 2^n * Sum_{k>=0} (k + 1/2)^n / k!. This is a Dobinski-type formula. - _Karol A. Penson_ and _Olivier Gérard_, Oct 22 2007
%F A126390 G.f.: 1/Q(0), where Q(k)= 1 - (2*k+3)*x - 4*(k+1)*x^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 03 2013
%F A126390 G.f.: 1/Q(0), where Q(k)= 1 - x - 2*x/(1 - 2*x*(2*k+1)/(1 - x - 2*x/(1 - 2*x*(2*k+2)/Q(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, May 13 2013
%F A126390 a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^k * a(n-k). - _Ilya Gutkovskiy_, Jun 21 2022
%F A126390 From _Vaclav Kotesovec_, Jun 22 2022: (Start)
%F A126390 a(n) ~ Bell(n) * (2 + LambertW(n)/n)^n.
%F A126390 a(n) ~ Bell(n) * 2^n * sqrt(n) * log(n)^(-1/2 + 1/(2*log(n)) - 1/(2*log(n)^2)) * exp(log(log(n))^2/(4*log(n)^2)). (End)
%F A126390 a(n) ~ 2^n * n^(n + 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/2)). - _Vaclav Kotesovec_, Jun 27 2022
%p A126390 with(combstruct):seq(count(([S, {N=Union(Z, S, P), S=Set(Union(Z, P), card>=0), P=Set(Union(Z, Z), card>=1)}, labeled], size=n)), n=0..20); # _Zerinvary Lajos_, Mar 18 2008
%t A126390 Table[ Sum[ 2^k Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 30}] (* _Karol A. Penson_ and _Olivier Gérard_, Oct 22 2007 *)
%o A126390 (PARI) x='x+O('x^66); Vec(serlaplace((exp(exp(2*x)-1+x)))) \\ _Joerg Arndt_, May 13 2013
%Y A126390 Cf. A000110, A000296, A005493, A124311, A126617, A284859, A285064.
%K A126390 nonn
%O A126390 0,2
%A A126390 _N. J. A. Sloane_, Aug 04 2007