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A271385 a(n) = Product_{k=0..floor((n - 1)/2)} (n - 2*k)^(n - 2*k).

%I A271385 #10 Feb 16 2025 08:33:33
%S A271385 1,1,4,27,1024,84375,47775744,69486440625,801543976648704,
%T A271385 26920470805806965625,8015439766487040000000000,
%U A271385 7680724499239438722449399746875,71466466094944065310414602240000000000,2326300251412874049290421829657963142035959375
%N A271385 a(n) = Product_{k=0..floor((n - 1)/2)} (n - 2*k)^(n - 2*k).
%C A271385 Double hyperfactorial (by analogy with the double factorial).
%H A271385 Ilya Gutkovskiy, <a href="/A271385/b271385.txt">Table of n, a(n) for n = 0..33</a>
%H A271385 Ilya Gutkovskiy, <a href="/A271385/a271385.pdf">Double hyperfactorial</a>
%H A271385 Eric Weisstein, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>
%H A271385 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>
%H A271385 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A271385 a(n) = n^n*(n - 2)^(n - 2)*...*5^5*3^3*1^1, for n>0 odd; a(n) = n^n*(n - 2)^(n - 2)*...*6^6*4^4*2^2, for n>0 even; a(n) = 1, for n = 0.
%F A271385 a(n) = n^n*a(n-2), a(0)=1, a(1)=1.
%F A271385 a(n) = (1/a(n-1))*sqrt(a(2n)/2^(n*(n+1))).
%F A271385 a(n)*a(n-1) = A002109(n).
%F A271385 a(n)*a(n-1)*sqrt(a(2n))/((n!)^n*sqrt(2^(n*(n+1)))) = A168510(n).
%e A271385 a(0) = 1;
%e A271385 a(1) = 1^1 = 1;
%e A271385 a(2) = 2^2 = 4;
%e A271385 a(3) = 1^1*3^3 = 27;
%e A271385 a(4) = 2^2*4^4 = 1024;
%e A271385 a(5) = 1^1*3^3*5^5 = 84375;
%e A271385 a(6) = 2^2*4^4*6^6 = 47775744;
%e A271385 a(7) = 1^1*3^3*5^5*7^7 = 69486440625;
%e A271385 a(8) = 2^2*4^4*6^6*8^8 = 801543976648704, etc.
%t A271385 Table[Product[(n - 2 k)^(n - 2 k), {k, 0, Floor[(n - 1)/2]}], {n, 0, 13}]
%t A271385 RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == n^n a[n - 2]}, a, {n, 13}]
%o A271385 (PARI) a(n) = prod(k=0, (n-1)\2, (n-2*k)^(n-2*k)); \\ _Michel Marcus_, Apr 07 2016
%Y A271385 Cf. A002109, A006882, A168510.
%K A271385 nonn,easy
%O A271385 0,3
%A A271385 _Ilya Gutkovskiy_, Apr 06 2016