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Double factorial
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(Redirected from Sextuple factorial)
n |
n |
-
n!! := n∏ i = 1
[·] |
n = 0 |
Alternatively, we have
|
The double factorial of nonnegative integers is defined recursively as
|
A006882 Double factorials
n!!: a (0) = a (1) = 1; a (n) = n ⋅ a (n − 2), n ≥ 2 |
- {1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, 185794560, 654729075, 3715891200, 13749310575, 81749606400, 316234143225, ...}
Contents
Generating functions for n!!
The generating function forn!! |
-
G{n!!}(x) := ∞∑ n = 0
n!! |
-
E{n!!}(x) := ∞∑ n = 0
= 1 + e x 2 / 2 (1 +x n n! √erf (π 2
)),x √ 2
where
-
erf (z) := 2 √ π
e − t 2d t =∫ z 0 2 √ π∞∑ n = 0(−1) n z 2 n + 1 n! (2 n + 1)
is the error function (erf ).[1]
A [generalized] continued fraction generating function forn!! |
C{n!!}(x) = ? +
|
Sum of reciprocals of double factorial of nonnegative integers
- ∞
∑ n = 0
=1 n!! ∞∑ n = 0
+1 (2 n)!!
} =1 (2 n + 1)!! ∞∑ n = 0
+1 (2 n)!! ∞∑ n = 0
=1 (2 n + 1)!! √ e+√ e∞∑ n = 0
=(−1) n (2 n)!! (2 n + 1) √ e{1 +∞∑ n = 0
}.(−1) n (2 n)!! (2 n + 1)
Double factorial of even nonnegative integers
The double factorial of even nonnegative integers is given by
-
(2 n)!! = 2 n n!, n ≥ 0.
(2 n)!! = 2 n n!, n ≥ 0 |
- {1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, ...}
Generating functions for (2 n)!!
The generating function for(2 n)!! |
-
G{(2 n)!!}(x) := ∞∑ n = 0
(2 n)!! |
-
E{(2 n)!!}(x) := ∞∑ n = 0
= E{2 n n!}(x) =x n n!
=1 1 − 2 x ∞∑ n = 0∞∑ n = 0
=x n n! ∞∑ n = 0
.x n n!
Note the following Maclaurin series expansion
- √ 1 + sin x=∞
∑ n = 0
x n = 1 +(−1) ⌊ n / 2⌋(2 n)!!
x −1 2
x 2 −1 8
x 3 +1 48
x 4 +1 384
x 5 −1 3840
x 6 −1 46080
x 7 + ⋯.1 645120
(2 n)!! |
C{(2 n)!!}(x) =
|
Sum of reciprocals of double factorial of even nonnegative integers
The sum of reciprocals of double factorial of even nonnegative integers equals√ e |
- ∞
∑ n = 0
=1 (2 n)!! ∞∑ n = 0
= {1 2 n n! ∞∑ n = 0
}x =x n n!
= {e x}x =1 2
=1 2 √ e.
Double factorial of odd nonnegative integers
The double factorial of odd nonnegative integers is given by
-
(2 n + 1)!! = (2 n + 1) (2 n − 1)!! = (2 n + 1)
=(2 n)! (2 n)!!
=(2 n + 1)! (2 n)!!
, n ≥ 0.(2 n + 1)! 2 n n!
(2 n − 1)!! = 1 ⋅ 3 ⋅ 5 ⋅ ... ⋅ (2 n − 1), n ≥ 1 |
- {1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, 13749310575, 316234143225, 7905853580625, 213458046676875, 6190283353629375, 191898783962510625, ...}
Generating functions for (2 n + 1)!!
The generating function for(2n + 1)!! |
-
G{(2 n + 1)!!}(x) := ∞∑ n = 0
(2n + 1)!! |
-
E{(2 n + 1)!!}(x) := ∞∑ n = 0
=x n n!
.1 √ 1 − 2 x
(2n + 1)!! |
C{(2 n + 1)!!}(x) =
|
Sum of reciprocals of double factorial of odd nonnegative integers
- ∞
∑ n = 0
=1 (2 n + 1)!! ∞∑ n = 0
=2 n n! (2 n + 1)! ∞∑ n = 0
=(2 n)!! (2 n)! (2 n + 1) √erf (π e 2
) =1 √ 2√ e∞∑ n = 0
=(−1) n 2 n n! (2 n + 1) √ e∞∑ n = 0
.(−1) n (2 n)!! (2 n + 1)
√ π e / 2 |
x = 1 |
√ 2 = 1.414213562373095... |
- ∞
∑ n = 0
= 1.410686134642447997690824711419115041323478....1 (2 n + 1)!!
|
- {1, 4, 1, 0, 6, 8, 6, 1, 3, 4, 6, 4, 2, 4, 4, 7, 9, 9, 7, 6, 9, 0, 8, 2, 4, 7, 1, 1, 4, 1, 9, 1, 1, 5, 0, 4, 1, 3, 2, 3, 4, 7, 8, 6, 2, 5, 6, 2, 5, 1, 9, 2, 1, 9, 7, 7, 2, 4, 6, 3, 9, 4, 6, 8, 1, 6, 4, 7, 8, 1, 7, 9, 8, 4, 9, 0, 3, 9, ...}
Double factorial binomial coefficients
The double factorial binomial coefficients are[2]
((
|
Multifactorial
Thek |
n |
(mod k ) |
n |
[·] |
n = 0 |
Alternatively, we have
The multifactorial of nonnegative integers is defined recursively as
|
A-number | ||
---|---|---|---|
0 | {1, 1, ?, ...} (Is it possible to generalize for k = 0?) | A?????? | |
1 | {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, ...} | A000142 | |
2 | {1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, 185794560, 654729075, 3715891200, 13749310575, 81749606400, ...} | A006882 | |
3 | {1, 1, 2, 3, 4, 10, 18, 28, 80, 162, 280, 880, 1944, 3640, 12320, 29160, 58240, 209440, 524880, 1106560, 4188800, 11022480, 24344320, 96342400, 264539520, 608608000, ...} | A007661 | |
4 | {1, 1, 2, 3, 4, 5, 12, 21, 32, 45, 120, 231, 384, 585, 1680, 3465, 6144, 9945, 30240, 65835, 122880, 208845, 665280, 1514205, 2949120, 5221125, 17297280, 40883535, 82575360, ...} | A007662 | |
5 | {1, 1, 2, 3, 4, 5, 6, 14, 24, 36, 50, 66, 168, 312, 504, 750, 1056, 2856, 5616, 9576, 15000, 22176, 62832, 129168, 229824, 375000, 576576, 1696464, 3616704, 6664896, ...} | A085157 | |
6 | {1, 1, 2, 3, 4, 5, 6, 7, 16, 27, 40, 55, 72, 91, 224, 405, 640, 935, 1296, 1729, 4480, 8505, 14080, 21505, 31104, 43225, 116480, 229635, 394240, 623645, 933120, 1339975, ...} | A085158 | |
7 | {1, 1, 2, 3, 4, 5, 6, 7, 8, 18, 30, 44, 60, 78, 98, 120, 288, 510, 792, 1140, 1560, 2058, 2640, 6624, 12240, 19800, 29640, 42120, 57624, 76560, 198720, 379440, 633600, 978120, ...} | A114799 | |
8 | {1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 33, 48, 65, 84, 105, 128, 153, 360, 627, 960, 1365, 1848, 2415, 3072, 3825, 9360, 16929, 26880, 39585, 55440, 74865, 98304, 126225, 318240, ...} | A114800 | |
9 | {1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 440, 756, 1144, 1610, 2160, 2800, 3536, 4374, 5320, 12760, 22680, 35464, 51520, 71280, 95200, ...} | A114806 | |
10 | |||
11 | |||
12 |
Multifactorial binomial coefficients
The multifactorial binomial coefficients are[2]
See also
Notes
- ↑ Weisstein, Eric W., Erf, from MathWorld—A Wolfram Web Resource.
- ↑ 2.0 2.1 http://www.maa.org/pubs/mag_jun12_toc.html