A337000 - OEIS

%I #7 Aug 09 2021 06:31:45

%S 1,2,7,34,211,1596,14259,147106,1722683,22591408,328161643,5232410646,

%T 90880495347,1708304787892,34557610000931,748647340991818,

%U 17294012137850443,424368973372593432,11024176183963527099,302260934814635426014,8722906492537092723395

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

%C Exponential convolution of Fubini numbers (A000670) and factorial numbers (A000142).

%F a(n) = n! * Sum_{k=0..n} A000670(k) / k!.

%F a(0) = 1; a(n) = 2 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (k-1) * a(n-k).

%F a(n) ~ n! / (2 * (1 - log(2)) * log(2)^(n+1)). - _Vaclav Kotesovec_, Aug 09 2021

%t nmax = 20; CoefficientList[Series[1/((1 - x) (2 - Exp[x])), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[n! Sum[HurwitzLerchPhi[1/2, -k, 0]/(2 k!), {k, 0, n}], {n, 0, 20}]

%t a[0] = 1; a[n_] := a[n] = 2 n a[n - 1] - Sum[Binomial[n, k]  (k - 1) a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 20}]

%Y Cf. A000142, A000670, A101053.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Aug 10 2020