A363842 -id:A363842 - cp's OEIS frontend

A363841 Continued fraction expansion of Sum_{k>=0} 1/(k!)!^2.

2, 3, 1, 32399, 4, 1432456210278611587930429493084159999, 1, 3, 32399, 1, 3

Offset: 0

Daniel Hoyt, Jun 23 2023

In general, sums of the form Sum_{k>=0} 1/(k!)!^t, t > 1 in N, have the following continued fraction expansion formulas:
The first term is always 2.
Let P(k) = (((k+1)!)! / ((k!)!)^2)^t - 1.
Take the sequence A157196 and replace the runs of '1,1' with 2^t - 1, the odd occurring runs of '2' with 2^t, and the even occurring runs of '2' with 2^t - 2. Finally interleave the modified sequence with a string of 1's and let it be called f(n). To get the continued fraction expansion, interleave the n-th runs of '2^t', '2^t - 1, 1', '1, 2^t - 1' and '1, 2^t - 2, 1' in f(n), and P(A001511(n)+1).
The next term a(11) has 303 digits. - Stefano Spezia, Jun 24 2023

Daniel Hoyt, Table of n, a(n) for n = 0..22
Daniel Hoyt, Python program that generates the continued fraction from formula.

Cf. A363842 (decimal expansion).

Cf. A001511, A336810.

ContinuedFraction[Sum[1/(k!)!^2, {k, 0, 6}], 21]

Take the sequence A157196 and replace the runs of '1,1' with '3'. Then replace the odd occurring runs of '2' with '4'. Finally interleave the modified sequence with a string of 1's and let it be called f(n). To get the continued fraction expansion, interleave between the n-th runs of '4', '3, 1', '1, 3' and '1, 2, 1' in f(n), and P(A001511(n)+1).

A387268 Decimal expansion of Sum_{k>=0} (-1)^k/(k!)!.

Original entry on oeis.org

4, 9, 8, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 2, 2, 8, 4, 8, 6, 8, 2, 2, 0, 7, 2, 2, 9, 4, 6, 0, 1, 5, 9, 8, 2, 4, 4, 1, 5, 9, 1, 2, 2, 8, 2, 9, 1, 2, 4, 3, 5, 8, 3, 7, 2, 1, 3, 7, 1, 8, 3, 3, 9, 0, 8, 4, 6, 4, 0, 4, 0, 1, 1, 4, 2, 1, 2, 1, 5, 6, 1, 6, 5, 9, 6, 3, 7, 9

Offset: 0

Daniel Hoyt, Aug 24 2025

This constant has an interesting simple continued fraction representation.

359/720 approximates this constant to 19 significant digits.

			0.498611111111111111127228486822072294...

Daniel Hoyt, Table of n, a(n) for n = 0..999

Cf. A386384 (continued fraction expansion).

Cf. A363842, A336810.

RealDigits[Sum[(-1)^k/(k!)!, {k, 0, 6}], 10, 100][[1]]

PARI
```
suminf(k=0, (-1)^k/(k!)!)
```

cp's OEIS Frontend

A363841 Continued fraction expansion of Sum_{k>=0} 1/(k!)!^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula