A386385 -id:A386385 - cp's OEIS frontend

A386384 Continued fraction expansion of Sum_{k>=0} (-1)^k/(k!)!.

0, 2, 179, 1, 1, 1196852626800230399, 2, 179, 1, 1, 17377308326435956818596067989554034737368967210468674554156131654360754429984573360106123813424835044026977477398690421454067571097599999999999999999999, 2, 179, 2, 1196852626800230399, 1, 1, 179, 2

Offset: 0

Daniel Hoyt, Aug 17 2025

The peak terms have the form P(k) = ((k+1)!)! / ((k!)!)^2 - 1. The sequence is an interleaving between the n-th runs of '2' and '1,1' in A386385, and P(A001511(n)+1).

Daniel Hoyt, Table of n, a(n) for n = 0..39
Daniel Hoyt, Justification for the explicit formula and program provided, Zenodo, 2025.

Cf. A001511, A157196, A336810, A386385.

Cf. A387268 (decimal expansion).

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

import sys #for printing huge factorials
sys.set_int_max_str_digits(0)  # otherwise sys not needed.
def a386384(n):
    import math
    if n==0: return 0
    t=n-1; M=0x18199818
    s=[1,1]; h=2; p=0; r=0
    def g(u):
        nonlocal s,h
        while len(s)>(r&31))&1) else ('A' if xb=='B' else 'B')
        L = 2 if yb=='A' else 1
        if t < L: return 1 if yb=='A' else 2
        t -= L
        if t==0:
            rr=r+1
            K=((rr & -rr).bit_length()-1)+2
            A=math.factorial(math.factorial(K+1))
            B=math.factorial(math.factorial(K))
            return A//(B*B)-1
        t-=1; r+=1

cp's OEIS Frontend

A386384 Continued fraction expansion of Sum_{k>=0} (-1)^k/(k!)!.

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