A143382 Numerator of Sum_{k=0..n} 1/k!!.

1, 2, 5, 17, 71, 121, 731, 1711, 41099, 370019, 740101, 2713789, 1206137, 423355111, 846710651, 1814380259, 203210595443, 12654139763, 531473870981, 43758015399281, 525096184837561, 441080795274037, 22054039763790029

Offset: 0

Author

Jonathan Vos Post, Aug 11 2008

Keywords

• easy
• frac
• nonn

Comments

Denominators are A143383. A143382(n)/A143383(n) is to A007676(n)/A007676(n) as double factorials are to factorials. A143382/A143383 fractions begin:

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

The series converges to sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) = 3.0594074053425761445... whose decimal expansion is given by A143280. The analogs of A094007 and A094008 are determined by 2 being the only prime denominator in the convergents to the sum of reciprocals of double factorials and prime numerators beginning: a(1) = 2, a(2) = 5, a(3) = 17, a(4) = 71, a(15) = 1814380259, a(19) = 43758015399281, a(21) = 441080795274037, a(23) = 867081905243923.

Examples

			a(3) = 17 because 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6.
a(15) = 1814380259 because 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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Eric W. Weisstein, Double Factorial. Gives formula for limit of series, which was independently derived by Max Alekseyev.

Crossrefs

Cf. A006882 (n!!), A094007, A143280 (m(2)), A143383 (denominators).

Programs

Magma

[n le 0 select 1 else Numerator( 1 + (&+[ 1/(0 + (&*[k-2*j: j in [0..Floor((k-1)/2)]])) : k in [1..n]]) ): n in [0..25]]; // G. C. Greubel, Mar 28 2019

Mathematica

Table[Numerator[Sum[1/k!!, {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Mar 28 2019 *)
Accumulate[1/Range[0,30]!!]//Numerator (* Harvey P. Dale, May 19 2023 *)

PARI

vector(25, n, n--; numerator(sum(k=0,n, 1/prod(j=0,floor((k-1)/2), (k - 2*j)) ))) \\ G. C. Greubel, Mar 28 2019

Sage

[numerator(sum( 1/product((k - 2*j) for j in (0..floor((k-1)/2)))   for k in (0..n))) for n in (0..25)] # G. C. Greubel, Mar 28 2019

Formula

Numerators of Sum_{k=0..n} 1/k!! = Sum_{k=0..n} 1/A006882(k).