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

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

Offset: 1

Author

Ilya Gutkovskiy, May 25 2022

Keywords

• nonn
• frac

Examples

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Python

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

Formula

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