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%I #14 Feb 20 2022 06:42:30
%S 1,1,10,102,1336,22200,443664,10334128,275060608,8236914048,
%T 274069953280,10031110907136,400520747437056,17324601073921024,
%U 807023462798608384,40278407730378332160,2144307919689898491904,121291661335680615284736,7264376142168665821741056
%N E.g.f.: 1 / (1 - x * exp(4*x)).
%H Seiichi Manyama, <a href="/A336952/b336952.txt">Table of n, a(n) for n = 0..375</a>
%F a(n) = n! * Sum_{k=0..n} (4 * (n-k))^k / k!.
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 4^(k-1) * a(n-k).
%F a(n) ~ n! * (4/LambertW(4))^n / (1 + LambertW(4)). - _Vaclav Kotesovec_, Aug 09 2021
%t nmax = 18; CoefficientList[Series[1/(1 - x Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t Join[{1}, Table[n! Sum[(4 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%o (PARI) seq(n)={ Vec(serlaplace(1 / (1 - x*exp(4*x + O(x^n))))) } \\ _Andrew Howroyd_, Aug 08 2020
%Y Column k=4 of A351790.
%Y Cf. A002697, A006153, A235328, A326324, A328183, A336950, A336951.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 08 2020
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