A336686 -id:A336686 - cp's OEIS frontend

A000197 a(n) = (n!)!.

1, 1, 2, 720, 620448401733239439360000

Offset: 0

The sequence 1, 2, 720!, 4!!!!, ... ,n!!...! (n times) grows too rapidly to have its own entry. See Hofstadter.
a(n) is divisible by 2^A245087(n) but not by 2^(A245087(n)+1), A245087 being the number of trailing zeros in its binary expansion. Also, for n>1, the largest prime divisor of a(n) is the largest prime <= n!, which is listed in A006990(n). - Stanislav Sykora, Jul 14 2014
See b-file for a(5), which has 199 digits and is too large to include. - Jianing Song, Jun 28 2018

Archimedeans Problems Drive, Eureka, 37 (1974), 11.
Douglas R. Hofstadter, Fluid concepts & creative analogies: computer models of the fundamental mechanisms of thought, Basic Books, 1995, pages 44-46. [From Colin Rowat, Sep 30 2011]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Franklin T. Adams-Watters, Table of n, a(n) for n=0..5
Rudolph Ondrejka, 1273 exact factorials, Math. Comp., 24 (1970), 231.
Eric Weisstein's World of Mathematics, Factorial.
Index entries for sequences related to factorial numbers

Cf. A063979. - Robert G. Wilson v, Dec 04 2008
Cf. A152168. - Alois P. Heinz, Aug 04 2013
Cf. A000142, A006990, A245087.

[Factorial(Factorial(n)) : n in [0..5]]; // Wesley Ivan Hurt, Jul 14 2014

A000197:=n->(n!)!: seq(A000197(n), n=0..5); # Wesley Ivan Hurt, Jul 14 2014

Table[(n!)!, {n, 0, 5}] (* Wesley Ivan Hurt, Jul 14 2014 *)

a(n) = A000142(A000142(n)). - Wesley Ivan Hurt, Jul 14 2014

Sum_{n>=0} 1/a(n) = A336686. - Amiram Eldar, Mar 10 2021

A336810 Continued fraction expansion of Sum_{k>=0} 1/(k!)!.

Original entry on oeis.org

2, 1, 1, 179, 2, 1196852626800230399, 1, 1, 179, 1, 1

Offset: 0

Daniel Hoyt, Nov 20 2020

a(11), a(21), and a(41) have 152, 1349, and 12981 digits, respectively.

The peak terms have the form ((k+1)!)! / ((k!)!)^2 - 1. Empirically the initial runs mimic an interleaving between the n-th runs of '1,1' and '2' in A157196, and P(A001511(n)+1) for a very long prefix. - Daniel Hoyt, Aug 29 2025

Georg Fischer, Table of n, a(n) for n = 0..20
Georg Fischer, Table of n, a(n) for n = 0..139
Alfred J. van der Poorten and Jeffrey Shallit, Folded continued fractions, Journal of Number Theory, Vol. 40, Issue 2, 1992, pp. 237-250 (cf. prop. 2).

Cf. A336686 (decimal expansion).

Cf. A001511, A157196, A363841.

ContinuedFraction[Sum[1/(k!)!, {k, 0, 6}], 21] (* Amiram Eldar, Nov 22 2020 *)

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

The peak terms have the form ((k+1)!)! / ((k!)!)^2 - 1. - Georg Fischer, Oct 19 2022 [pers. comm. with J. Shallit]

A363842 Decimal expansion of Sum_{k>=0} 1/(k!)!^2.

Original entry on oeis.org

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

Offset: 1

Daniel Hoyt, Jun 23 2023

1166401/518400 approximates this constant to 48 significant digits.

			2.25000192901234567901234567901234567901234567901...

Cf. A363841 (continued fraction).

Cf. A336686.

RealDigits[Sum[1/(k!)!^2, {k, 0, 4}], 10, 100][[1]]

PARI
```
suminf(k=0, 1/k!!^2)
```

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A000197 a(n) = (n!)!.

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A336810 Continued fraction expansion of Sum_{k>=0} 1/(k!)!.

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A363842 Decimal expansion of Sum_{k>=0} 1/(k!)!^2.

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