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.

```