A354402 -id:A354402 - cp's OEIS frontend

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

Ilya Gutkovskiy, May 25 2022

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

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

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

a(n) = numerator(sum(k=1, n, 1/(k*k!))); \\ Michel Marcus, May 26 2022

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

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

A354404 a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

Original entry on oeis.org

1, 4, 36, 288, 7200, 10800, 264600, 33868800, 914457600, 4572288000, 553246848000, 2212987392000, 373994869248000, 327245510592000, 19634730635520000, 5026491042693120000, 1452655911338311680000, 39221709606134415360000, 14159037167814523944960000, 141590371678145239449600000

Offset: 1

Ilya Gutkovskiy, May 25 2022

			1, 3/4, 29/36, 229/288, 5737/7200, 8603/10800, 210781/264600, ...

Cf. A001563, A053556, A061355, A239069, A354401, A354402 (numerators).

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

a(n) = denominator(sum(k=1, n, (-1)^(k+1)/(k*k!))); \\ Michel Marcus, May 26 2022

from math import factorial
from fractions import Fraction
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

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

cp's OEIS Frontend

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

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