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 A334190 #8 Apr 18 2020 11:49:38
%S A334190 1,0,-2,-4,4,64,248,48,-6512,-51200,-171296,830400,17870400,144684032,
%T A334190 441316224,-5976726784,-119879356160,-1123892297728,-3962230563328,
%U A334190 70410917051392,1686366492509184,19578100126072832,101728414306826240,-1258662784047370240,-42727186269262737408
%N A334190 a(n) = exp(1/2) * Sum_{k>=0} (2*k + 1)^n / ((-2)^k * k!).
%F A334190 G.f.: (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - 2*j*x/(1 - x)).
%F A334190 E.g.f.: exp(x + (1 - exp(2*x)) / 2).
%t A334190 nmax = 24; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 2 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A334190 nmax = 24; CoefficientList[Series[Exp[x + (1 - Exp[2 x])/2], {x, 0, nmax}], x] Range[0, nmax]!
%t A334190 Table[Sum[Binomial[n, k] * 2^k * BellB[k, -1/2], {k, 0, n}], {n, 0, 24}] (* _Vaclav Kotesovec_, Apr 18 2020 *)
%Y A334190 Column k=2 of A334192.
%Y A334190 Cf. A007405, A009235, A293037, A308536, A334191.
%K A334190 sign
%O A334190 0,3
%A A334190 _Ilya Gutkovskiy_, Apr 18 2020