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A337354 a(n) is the numerator of Product_{i=0..n-1} (n-i)^((-1)^ceiling(i/2)).

%I A337354 #69 Oct 06 2020 13:09:46
%S A337354 1,2,3,2,5,9,7,40,45,7,308,48,975,539,88,1664,1105,24255,13376,56576,
%T A337354 41769,48279,55936,226304,348075,370139,671232,870400,2082925,4283037,
%U A337354 13872128,80773120,343682625,4023459,1553678336,1900544,14411758075,59457783,1471905792,1406402560
%N A337354 a(n) is the numerator of Product_{i=0..n-1} (n-i)^((-1)^ceiling(i/2)).
%C A337354 a(n) is the numerator of (n/(n-1)) * ((n-3)/(n-2)) * ((n-4)/(n-5)) ...
%F A337354 a(n) = numerator of (n*A337355(n-2))/(a(n-2)*(n-1)) for n>=3.
%F A337354 Conjecture: a(4*n)/A337355(4*n) ~ 0.5990701173677... (=A076390). - _Andrew Howroyd_, Aug 25 2020
%e A337354 a(n)/A337355(n) equals 1, 2, 3/2, 2/3, 5/6, 9/5, 7/5, 40/63, 45/56, 7/4 ...
%e A337354 a(4) = numerator of (4*1)/(3*2) = numerator of 2/3 = 2.
%e A337354 a(5) = numerator of (5*2)/(4*3) = numerator of 5/6 = 5.
%e A337354                       12  *   9*8  *  5*4  *  1
%e A337354 a(12) = numerator of --------------------------- = 48.
%e A337354                         11*10  *  7*6  *  3*2
%o A337354 (PARI) a(n) = {numerator(prod(i=0, n-1, (n-i)^(-1)^((i+1)\2)))} \\ _Andrew Howroyd_, Aug 24 2020
%Y A337354 Cf. A337355 (denominators).
%Y A337354 Cf. A076390, A085565, A007662.
%K A337354 nonn,frac
%O A337354 1,2
%A A337354 _Devansh Singh_, Aug 24 2020
%E A337354 Terms a(31) and beyond from _Andrew Howroyd_, Aug 25 2020