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a(n) = n! * (1 + Sum_{k=0..n} binomial(k+3,4) / k!).
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%I #10 Jan 05 2024 07:56:43

%S 1,2,9,42,203,1085,6636,46662,373626,3363129,33632005,369953056,

%T 4439438037,57712696301,807977750594,12119666261970,193914660195396,

%U 3296549223326577,59337886019884371,1127419834377810364,22548396687556216135,473516330438680549461

%N a(n) = n! * (1 + Sum_{k=0..n} binomial(k+3,4) / k!).

%F a(0) = 1; a(n) = n*a(n-1) + binomial(n+3,4).

%F a(n) = n! + A368575(n).

%F E.g.f.: (1 + x * (1+3*x/2+x^2/2+x^3/24) * exp(x)) / (1-x).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x*sum(k=0, 3, binomial(3, k)*x^k/(k+1)!)*exp(x))/(1-x)))

%Y Cf. A033540, A368762, A368763.

%Y Cf. A368575, A368768.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 04 2024