SEPHER SEPHIROTH

SEPHER SEPHIROTH
3283 7 3293 37 3301 P. 3309 — 3317 31
3287 19 3297 — 3303 — 3311 7 3319 P.
3289 11 3299 P. 3307 P. 3313 P. 3321 —
3291 —
The first dozen factorials, and sub-factorials; and the ratios they bear to
one another; note that |n / ||n = e
xv
N |N ||N |N ÷ ||N ||N ÷ |N
1
2
3
4
5
6
7
8
9
10
11
12
1
2
6
24
120
720
5040
40320
362880
2628800
39916800
479001600
0
1
2
9
44
265
1854
14833
133496
1334961
14684570
176214841
∞
2.000000
3.000000
2.666666
2.727272
2.716981
2.718446
2.718262
2.718283
2.718281
2.718281
2.718281
0.000000
0.500000
0.333333
0.375000
0.366666
0.368055
0.367857
0.367881
0.367879
0.367879
0.367879
0.367879
Factorial n, or ƒn is the continued product of all the whole numbers from 1 to n inclusive and is
the number of ways in which n different things can be arranged.
Sub-factorial n, or ||n, is the nearest whole number to n ÷ e, and is the number of ways in which a
row of n elements may be so deranged, that no element may have its original position.
Thus ƒn = 1 × 2 × 3 × . . . × n,
1×
2×
3×
... × n
and ||n = ± h ,
2.
71828188...
1×
2×
3×
... × n
where h is the smaller decimal fraction less than unity by which the fraction
differs from a
2.
71828188...
whole number, and is to be added or subtracted as the case may be.—The most useful expression for
||n is:
||n ≡ n n(
n − 1)
n(
n − 1)(
n − 2)
etc
n! − ( n − 1)!
+ ( n − 2)!
−
( n − 3)!
+
1 1⋅2
1⋅2
⋅3
to (n+1) terms.
1 1 1
e ≡ 1 + + + + ... to ∞
1!
2!
3!
≡ 2.71828188... .