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a(n) = n! * (1 + Sum_{k=0..n} binomial(k+1,2) / k!).
3

%I #14 Jan 05 2024 07:53:54

%S 1,2,7,27,118,605,3651,25585,204716,1842489,18424945,202674461,

%T 2432093610,31617217021,442641038399,6639615576105,106233849217816,

%U 1805975436703025,32507557860654621,617643599352437989,12352871987048759990,259410311728023960021

%N a(n) = n! * (1 + Sum_{k=0..n} binomial(k+1,2) / k!).

%F a(0) = 1; a(n) = n*a(n-1) + binomial(n+1,2).

%F a(n) = n! + A103519(n).

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x*sum(k=0, 1, binomial(1, k)*x^k/(k+1)!)*exp(x))/(1-x)))

%Y Cf. A033540, A368763, A368764.

%Y Cf. A103519, A368766.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 04 2024