A353545 -id:A353545 - cp's OEIS frontend

A354401 a(n) is the denominator of Sum_{k=1..n} 1 / (k*k!).

1, 4, 36, 288, 7200, 10800, 66150, 33868800, 914457600, 4572288000, 553246848000, 737662464000, 41554985472000, 54540918432000, 19634730635520000, 5026491042693120000, 1452655911338311680000, 39221709606134415360000, 14159037167814523944960000, 141590371678145239449600000

Offset: 1

Ilya Gutkovskiy, May 25 2022

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

Cf. A001563, A053556, A061355, A229837, A353545 (numerators), A354404.

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

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

from math import factorial
from fractions import Fraction
def A354401(n): return sum(Fraction(1, k*factorial(k)) for k in range(1,n+1)).denominator # Chai Wah Wu, May 27 2022

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

A354402 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (k*k!).

Original entry on oeis.org

1, 3, 29, 229, 5737, 8603, 210781, 26979863, 728456581, 3642282779, 440716217519, 1762864869691, 297924162982399, 260683642609331, 15641018556560861, 4004100750479565401, 1157185116888594641129, 31243998155992054970143, 11279083334313131850347743, 112790833343131318500567523

Offset: 1

Ilya Gutkovskiy, May 25 2022

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

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

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

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

from math import factorial
from fractions import Fraction
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

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

cp's OEIS Frontend

A354401 a(n) is the denominator of Sum_{k=1..n} 1 / (k*k!).

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