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A309087 a(n) = Sum_{k >= 0} floor(n^k / k!).

%I A309087 #9 Jul 14 2019 06:26:19
%S A309087 1,2,6,18,50,143,397,1088,2973,8093,22014,59861,162742,442396,1202589,
%T A309087 3268996,8886090,24154933,65659949,178482278,485165168,1318815708,
%U A309087 3584912818,9744803414,26489122097,72004899306,195729609397,532048240570,1446257064252
%N A309087 a(n) = Sum_{k >= 0} floor(n^k / k!).
%C A309087 This sequence is inspired by the Maclaurin series for the exponential function.
%C A309087 The series in the name is well defined; for any n > 0, only the first A065027(n) terms are different from zero.
%H A309087 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taylor_series#Exponential_function">Taylor series: Exponential function</a>
%F A309087 a(n) ~ exp(n) as n tends to infinity.
%F A309087 a(n) <= A000149(n).
%F A309087 a(n) = A309104(n) + A309105(n).
%e A309087 For n = 3:
%e A309087 - we have:
%e A309087   k  floor(3^k / k!)
%e A309087   -  ---------------
%e A309087   0                1
%e A309087   1                3
%e A309087   2                4
%e A309087   3                4
%e A309087   4                3
%e A309087   5                2
%e A309087   6                1
%e A309087   >=7              0
%e A309087 - hence a(3) = 1 + 3 + 4 + 4 + 3 + 2 + 1 = 18.
%o A309087 (PARI) a(n) = { my (v=0, d=1); for (k=1, oo, if (d<1, return (v), v += floor(d); d *= n/k)) }
%Y A309087 See A309103, A309104, A309105 for similar sequences.
%Y A309087 Cf. A000149, A065027.
%K A309087 nonn
%O A309087 0,2
%A A309087 _Rémy Sigrist_, Jul 11 2019