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E.g.f.: exp( exp(x) * (1 + x + x^2 / 2) - 1 ).
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%I #5 Sep 10 2021 19:22:47

%S 1,2,8,39,227,1518,11368,93796,842416,8158942,84581560,932878169,

%T 10891741957,134043979644,1732583270218,23445954950207,

%U 331260511278659,4874617929283392,74548457001207068,1182551615010825076,19423368875596930596,329809489306236629874

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

%C Exponential transform of A000124.

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000124(k) * a(n-k).

%t nmax = 21; CoefficientList[Series[Exp[Exp[x] (1 + x + x^2/2) - 1], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (k (k + 1)/2 + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

%Y Cf. A000085, A000124, A209801, A279361, A347666.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Sep 10 2021