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a(n) = exp(-1/5) * Sum_{k>=0} (5*k + 4)^n / (5^k * k!).
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%I #8 Jun 27 2023 11:44:52

%S 1,5,30,225,2075,22500,276875,3790625,57050000,934984375,16549046875,

%T 314146406250,6358972578125,136603266015625,3101556258593750,

%U 74164388642578125,1861859526474609375,48936176077929687500,1343302192888037109375,38425435693841064453125,1143143051078878906250000

%N a(n) = exp(-1/5) * Sum_{k>=0} (5*k + 4)^n / (5^k * k!).

%H Michael De Vlieger, <a href="/A363908/b363908.txt">Table of n, a(n) for n = 0..450</a>

%H Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, <a href="https://arxiv.org/abs/2306.13034">Flattened Stirling Permutations</a>, arXiv:2306.13034 [math.CO], 2023. See p. 14.

%t CoefficientList[Series[Exp[4 x + (Exp[5 x] - 1)/5], {x, 0, #}], x]* Range[0, #]! &[21]

%Y Cf. A007405, A050488, A355164, A355167.

%K nonn

%O 0,2

%A _Michael De Vlieger_, Jun 27 2023