A353545 a(n) is the numerator of Sum_{k=1..n} 1 / (k*k!).

1, 5, 47, 379, 9487, 14233, 87179, 44635753, 1205165611, 6025828181, 729125211161, 972166948343, 54765404757169, 71879593743829, 25876653747779441, 6624423359431551911, 1914458350875718742519, 51690375473644406388353, 18660225545985630712321553, 186602255459856307126125437

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Formula

Numerators of coefficients in expansion of (Ei(x) - log(x) - gamma) / (1 - x), x > 0.

Example

1, 5/4, 47/36, 379/288, 9487/7200, 14233/10800, 87179/66150, ...

Mathematica

Table[Sum[1/(k k!), {k, 1, n}], {n, 1, 20}] // Numerator
nmax = 20; Assuming[x > 0, CoefficientList[Series[(ExpIntegralEi[x] - Log[x] - EulerGamma)/(1 - x), {x, 0, nmax}], x]] // Numerator // Rest

Prog

(PARI) a(n) = numerator(sum(k=1, n, 1/(k*k!))); \\ Michel Marcus, May 26 2022
(Python)
from math import factorial
from fractions import Fraction
def A353545(n): return sum(Fraction(1, k*factorial(k)) for k in range(1, n+1)).numerator # Chai Wah Wu, May 27 2022

Crossrefs

Cf. A001563, A053557, A061354, A103816, A120265, A229837, A354401 (denominators), A354402.

Sequence in context: A354894 A219073 A327595 * A198854 A200814 A181582

Adjacent sequences: A353542 A353543 A353544 * A353546 A353547 A353548

Keyword

nonn,frac

Author

Ilya Gutkovskiy, May 25 2022

Status

approved