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Denominator of Sum_{k=0..n} 1/k!!.
9

%I #13 Sep 08 2022 08:45:37

%S 1,1,2,6,24,40,240,560,13440,120960,241920,887040,394240,138378240,

%T 276756480,593049600,66421555200,4136140800,173717913600,

%U 14302774886400,171633298636800,144171970854912,7208598542745600,283414985441280

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

%C Numerators are A143382. 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="/A143383/b143383.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 Denominators of Sum_{k=0..n} 1/k!! = Sum_{k=0..n} 1/A006882(k).

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

%e a(15) = 593049600 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[Denominator[Sum[1/k!!, {k,0,n}]], {n,0,25}] (* _G. C. Greubel_, Mar 28 2019 *)

%o (PARI) vector(25, n, n--; denominator(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 Denominator( 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) [denominator(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)), A143382 (numerator).

%K easy,frac,nonn

%O 0,3

%A _Jonathan Vos Post_, Aug 11 2008