A334190 - OEIS

%I #8 Apr 18 2020 11:49:38

%S 1,0,-2,-4,4,64,248,48,-6512,-51200,-171296,830400,17870400,144684032,

%T 441316224,-5976726784,-119879356160,-1123892297728,-3962230563328,

%U 70410917051392,1686366492509184,19578100126072832,101728414306826240,-1258662784047370240,-42727186269262737408

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

%F G.f.: (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - 2*j*x/(1 - x)).

%F E.g.f.: exp(x + (1 - exp(2*x)) / 2).

%t 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 nmax = 24; CoefficientList[Series[Exp[x + (1 - Exp[2 x])/2], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Binomial[n, k] * 2^k * BellB[k, -1/2], {k, 0, n}], {n, 0, 24}] (* _Vaclav Kotesovec_, Apr 18 2020 *)

%Y Column k=2 of A334192.

%Y Cf. A007405, A009235, A293037, A308536, A334191.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Apr 18 2020