A354402 - OEIS

%I #15 May 27 2022 21:14:23

%S 1,3,29,229,5737,8603,210781,26979863,728456581,3642282779,

%T 440716217519,1762864869691,297924162982399,260683642609331,

%U 15641018556560861,4004100750479565401,1157185116888594641129,31243998155992054970143,11279083334313131850347743,112790833343131318500567523

%N a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

%F Numerators 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}] // Numerator

%t nmax = 20; Assuming[x > 0, CoefficientList[Series[(EulerGamma + Log[x] - ExpIntegralEi[-x])/(1 - x), {x, 0, nmax}], x]] // Numerator // Rest

%o (PARI) a(n) = numerator(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 A354402(n): return sum(Fraction(1 if k & 1 else -1, k*factorial(k)) for k in range(1,n+1)).numerator # _Chai Wah Wu_, May 27 2022

%Y Cf. A001563, A053557, A061354, A103816, A120265, A239069, A353545, A354404 (denominators).

%K nonn,frac

%O 1,2

%A _Ilya Gutkovskiy_, May 25 2022