A143382 - OEIS

%I #17 Oct 01 2024 23:19:12

%S 1,2,5,17,71,121,731,1711,41099,370019,740101,2713789,1206137,

%T 423355111,846710651,1814380259,203210595443,12654139763,531473870981,

%U 43758015399281,525096184837561,441080795274037,22054039763790029

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

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

%C n numerator/denominator

%C 0 1/0!! = 1/1

%C 1 1/0!! + 1/1!! = 2/1

%C 2 1/0!! + 1/1!! + 1/2!! = 5/2

%C 3 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6

%C 4 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! = 71/24

%C 5 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! = 121/40

%C 6 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! = 731/240

%C 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.

%H G. C. Greubel, <a href="/A143382/b143382.txt">Table of n, a(n) for n = 0..500</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>. Gives formula for limit of series, which was independently derived by Max Alekseyev.

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

%e a(3) = 17 because 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6.

%e 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.

%t Table[Numerator[Sum[1/k!!, {k, 0, n}]], {n, 0, 30}] (* _G. C. Greubel_, Mar 28 2019 *)

%t Accumulate[1/Range[0,30]!!]//Numerator (* _Harvey P. Dale_, May 19 2023 *)

%o (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

%o (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

%o (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

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

%K easy,frac,nonn

%O 0,2

%A _Jonathan Vos Post_, Aug 11 2008