A308946 - OEIS

%I #7 Aug 09 2021 03:21:26

%S 1,1,5,30,244,2485,30351,432502,7043660,129050649,2627117875,

%T 58829021416,1437117395946,38032508860177,1083932872119839,

%U 33098858988564090,1078083456543449416,37309607437056658129,1367138649165397662627,52879280631976735387588

%N Expansion of e.g.f. 1/(1 - x*(1 + x/2)*exp(x)).

%F E.g.f.: 1 / (1 - Sum_{k>=1} (k*(k + 1)/2)*x^k/k!).

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000217(k) * a(n-k).

%F a(n) ~ n! * (2 + r) / ((2 + 4*r + r^2) * r^n), where r = 0.49122518354447387971550543450091640839121607... is the root of the equation  exp(r)*r*(2 + r) = 2. - _Vaclav Kotesovec_, Aug 09 2021

%t nmax = 19; CoefficientList[Series[1/(1 - x (1 + x/2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k + 1, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

%Y Cf. A000217, A001793, A006153, A052529, A279361, A308861.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 02 2019