This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143383 #14 Feb 16 2025 08:33:08
%S A143383 1,1,2,6,24,40,240,560,13440,120960,241920,887040,394240,138378240,
%T A143383 276756480,593049600,66421555200,4136140800,173717913600,
%U A143383 14302774886400,171633298636800,144171970854912,7208598542745600,283414985441280
%N A143383 Denominator of Sum_{k=0..n} 1/k!!.
%C A143383 Numerators are A143382. A143382(n)/A143383(n) is to A007676(n)/A007676(n) as double factorials are to factorials. A143382/A143383 fractions begin:
%C A143383 n numerator/denominator
%C A143383 0 1/0!! = 1/1
%C A143383 1 1/0!! + 1/1!! = 2/1
%C A143383 2 1/0!! + 1/1!! + 1/2!! = 5/2
%C A143383 3 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6
%C A143383 4 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! = 71/24
%C A143383 5 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! = 121/40
%C A143383 6 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! = 731/240
%C A143383 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 A143383 G. C. Greubel, <a href="/A143383/b143383.txt">Table of n, a(n) for n = 0..500</a>
%H A143383 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>. Gives formula for limit of series, which was independently derived by Max Alekseyev.
%F A143383 Denominators of Sum_{k=0..n} 1/k!! = Sum_{k=0..n} 1/A006882(k).
%e A143383 a(3) = 6 because 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6.
%e A143383 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 A143383 Table[Denominator[Sum[1/k!!, {k,0,n}]], {n,0,25}] (* _G. C. Greubel_, Mar 28 2019 *)
%o A143383 (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 A143383 (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 A143383 (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 A143383 Cf. A006882 (n!!), A094007, A143280 (m(2)), A143382 (numerator).
%K A143383 easy,frac,nonn
%O A143383 0,3
%A A143383 _Jonathan Vos Post_, Aug 11 2008