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a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.
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%I #10 Jan 10 2024 23:55:50

%S 1,3,15,105,189,10395,135135,2027025,34459425,130945815,13749310575,

%T 316234143225,7905853580625,12556355686875,1238056670725875,

%U 776918153694375,6332659870762850625,7642865361265509375,8200794532637891559375,63966197354575554163125,13113070457687988603440625

%N a(n) is the denominator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

%H Robert Israel, <a href="/A354299/b354299.txt">Table of n, a(n) for n = 1..403</a>

%F Denominators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).

%e 1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...

%p S:= 0: R:= NULL:

%p for n from 1 to 100 do

%p S:= S + (-1)^(n+1)/doublefactorial(2*n-1);

%p R:= R, denom(S);

%p od:

%p R; # _Robert Israel_, Jan 10 2024

%t Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Denominator

%t nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Denominator // Rest

%t Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Denominator

%Y Cf. A001147, A053556, A061355, A064647, A113013, A143383, A289488, A306858, A354298 (numerators).

%K nonn,frac

%O 1,2

%A _Ilya Gutkovskiy_, May 23 2022