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%I #14 May 27 2022 21:14:31
%S 1,4,36,288,7200,10800,264600,33868800,914457600,4572288000,
%T 553246848000,2212987392000,373994869248000,327245510592000,
%U 19634730635520000,5026491042693120000,1452655911338311680000,39221709606134415360000,14159037167814523944960000,141590371678145239449600000
%N a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).
%F Denominators of coefficients in expansion of (gamma + log(x) - Ei(-x)) / (1 - x), x > 0.
%e 1, 3/4, 29/36, 229/288, 5737/7200, 8603/10800, 210781/264600, ...
%t Table[Sum[(-1)^(k + 1)/(k k!), {k, 1, n}], {n, 1, 20}] // Denominator
%t nmax = 20; Assuming[x > 0, CoefficientList[Series[(EulerGamma + Log[x] - ExpIntegralEi[-x])/(1 - x), {x, 0, nmax}], x]] // Denominator // Rest
%o (PARI) a(n) = denominator(sum(k=1, n, (-1)^(k+1)/(k*k!))); \\ _Michel Marcus_, May 26 2022
%o (Python)
%o from math import factorial
%o from fractions import Fraction
%o def A354404(n): return sum(Fraction(1 if k & 1 else -1, k*factorial(k)) for k in range(1,n+1)).denominator # _Chai Wah Wu_, May 27 2022
%Y Cf. A001563, A053556, A061355, A239069, A354401, A354402 (numerators).
%K nonn,frac
%O 1,2
%A _Ilya Gutkovskiy_, May 25 2022