Sum of reciprocals of double factorials
Jonathan Post
jvospost3 at gmail.com
Sat Aug 9 22:54:14 CEST 2008
Sum of reciprocals of double factorials
(2 seqs "frac" for numerators and denominators; shortly to be subitted to OEIS)
SUM[i=0..n] 1/A006882(i)
n numerator/denominator
0 1/0!! = 1/1
1 1/0!! + 1/1!! = 2/1
2 1/0!! + 1/1!! + 1/2!! = 5/2
3 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6
4 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! = 71/24
5 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! = 121/40
6 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! = 731/240
7 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! + 1/7!! =
1711/560 ~ 3.0553571
8 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! + 1/7!! +
1/8!! = 41099/13440 ~ 3.05796131
9 1/1+ 1/1+ 1/2 + 1/3 + 1/8 + 1/15 + 1/48 + 1/105 + 1/384 + 1/945:
= 370019/120960
10 740101/241920
11 2713789/887040
12 1206137/394240
13 423355111/138378240
14 846710651/276756480
15 1814380259/593049600 = 1/1 + 1/1 + 1/2 + 1/3 + 1/8 + 1/15 + 1/48 +
1/105 + 1/384 + 1/945 + 1/3840 + 1/10395 + 1/46080 + 1/135135 +
1/645120 + 1/2027025
16 203210595443/66421555200
17 12654139763/4136140800
18 531473870981/173717913600
19 43758015399281/14302774886400
20 525096184837561/171633298636800
21 441080795274037/144171970854912
22 22054039763790029/7208598542745600
23 867081905243923/283414985441280
24 312149485887864283/102029394758860800
25 101448582913560086279/33159553296629760000
= 1/1+ 1/1+ 1/2 + 1/3 + 1/8 + 1/15 + 1/48 + 1/105 + 1/384 + 1/945 +
1/3840 + 1/10395 + 1/46080 + 1/135135 + 1/645120 + 1/2027025 +
1/10321920 + 1/34459425 + 1/185794560 + 1/654729075 +1/3715891200 +
1/13749310575 + 1/81749606400 + 1/316234143225 + 1/1961990553600 +
1/7905853580625:
= 3 + 1/16+ 1/1+ 1/4+ 1/1+ 1/66+ 1/10+ 1/1+ 1/1+ 1/1+ 1/1+ 1/2+ 1/5+
1/352+ 1/1+ 1/2+ 1/1+ 1/2+ 1/2+ 1/1+ 1/23+ 1/2+ 1/4+ 1/1+ 1/1+ 1/3+
1/2+ 1/2+ 1/27+ 1/138+ 1/14 [this last expression is a continued
fraction]
The series converges to [Wesstein's seq]
This is to double factorials A006882 as A007676/A007677is to factorial.
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