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%I #13 May 28 2022 02:11:52
%S 1,4,36,288,7200,10800,66150,33868800,914457600,4572288000,
%T 553246848000,737662464000,41554985472000,54540918432000,
%U 19634730635520000,5026491042693120000,1452655911338311680000,39221709606134415360000,14159037167814523944960000,141590371678145239449600000
%N a(n) is the denominator of Sum_{k=1..n} 1 / (k*k!).
%F Denominators of coefficients in expansion of (Ei(x) - log(x) - gamma) / (1 - x), x > 0.
%e 1, 5/4, 47/36, 379/288, 9487/7200, 14233/10800, 87179/66150, ...
%t Table[Sum[1/(k k!), {k, 1, n}], {n, 1, 20}] // Denominator
%t nmax = 20; Assuming[x > 0, CoefficientList[Series[(ExpIntegralEi[x] - Log[x] - EulerGamma)/(1 - x), {x, 0, nmax}], x]] // Denominator // Rest
%o (PARI) a(n) = denominator(sum(k=1, n, 1/(k*k!))); \\ _Michel Marcus_, May 26 2022
%o (Python)
%o from math import factorial
%o from fractions import Fraction
%o def A354401(n): return sum(Fraction(1, k*factorial(k)) for k in range(1,n+1)).denominator # _Chai Wah Wu_, May 27 2022
%Y Cf. A001563, A053556, A061355, A229837, A353545 (numerators), A354404.
%K nonn,frac
%O 1,2
%A _Ilya Gutkovskiy_, May 25 2022