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A240988 Denominators of the (reduced) rationals (((n-1)!!)/(n!! * 2^((1 + (-1)^n)/2)))^((-1)^n), where n is a positive integer.

%I A240988 #35 Nov 09 2024 00:42:08
%S A240988 1,4,2,16,8,32,16,256,128,512,256,2048,1024,4096,2048,65536,32768,
%T A240988 131072,65536,524288,262144,1048576,524288,8388608,4194304,16777216,
%U A240988 8388608,67108864,33554432,134217728,67108864,4294967296,2147483648,8589934592,4294967296
%N A240988 Denominators of the (reduced) rationals (((n-1)!!)/(n!! * 2^((1 + (-1)^n)/2)))^((-1)^n), where n is a positive integer.
%C A240988 Numerators for this sequence are the swinging factorial A163590, starting from n = 1.
%C A240988 The terms are all powers of 2 (A000079).
%C A240988 It appears that a(2*n) = 2^A101925(n) and a(2*n+1) = 2^A005187(n). - _Robert Israel_, Aug 06 2014
%H A240988 James Burling, <a href="/A240988/a240988.pdf">The Special Rational Sequence</a>
%F A240988 a(n) = denominator((((n-1)!!)/(n!! * 2^((1 + (-1)^n)/2)))^((-1)^n)).
%F A240988 a(n) = denominator(g(1, n)) where g(m, n) = m if m = n; m/(2 * g(m + 1, n)) otherwise.
%e A240988 For n = 1, a(1) = 1.
%e A240988 For n = 2, a(2) = 2 * 2 = 4.
%e A240988 For n = 6, a(6) = 2 * 2 * 4 * 2 = 32.
%p A240988 f:= n -> denom(((doublefactorial(n-1)) / (doublefactorial(n)*2^((1+(-1)^n)/2)))^((-1)^n)):
%p A240988 seq(f(n), n=1..100); # _Robert Israel_, Aug 06 2014
%o A240988 (PARI)
%o A240988 df(n) = prod(i=0, floor((n-1)/2), n-2*i) \\ Double factorial (n!!)
%o A240988 a(n) = denominator(((df(n-1)) / (df(n)*2^((1+(-1)^n)/2)))^((-1)^n))
%o A240988 vector(50, n, a(n)) \\ _Colin Barker_, Aug 06 2014
%Y A240988 Cf. A163590 (numerators).
%K A240988 frac,nonn
%O A240988 1,2
%A A240988 _James Burling_, Aug 06 2014
%E A240988 More terms from _Colin Barker_, Aug 06 2014