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a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+1,2) / k!).
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%I #12 Jan 05 2024 07:57:09

%S 1,0,3,3,22,95,591,4109,32908,296127,2961325,32574509,390894186,

%T 5081624327,71142740683,1067141110125,17074257762136,290262381956159,

%U 5224722875211033,99269734629009437,1985394692580188950,41693288544183967719,917252347972047290071

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

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

%F a(n) = n! + (-1)^n * A009574(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. A368765, A368767, A368768.

%Y Cf. A009574, A368762.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 04 2024