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a(n) = ([n]_phi! - [n]_{1-phi}!)/sqrt(5), where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.
2

%I #24 Sep 24 2016 16:11:38

%S 0,0,1,6,58,948,25992,1179016,87713040,10646068080,2101395344400,

%T 673242645670320,349671381118477440,294206779308703578240,

%U 400822226102433353285760,883965927408694948620295680,3155212287401150653204012531200

%N a(n) = ([n]_phi! - [n]_{1-phi}!)/sqrt(5), where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>.

%F [n]_phi! = (A274983(n) + a(n)*sqrt(5))/2.

%F [n]_{1-phi}! = (A274983(n) - a(n)*sqrt(5))/2.

%F a(n) ~ c * phi^(n*(n+3)/2) / sqrt(5), where c = QPochhammer(phi-1) = A276987 = 0.1208019218617061294237231569887920563043992516794... . - _Vaclav Kotesovec_, Sep 24 2016

%e For n = 3, [3]_phi! = 1060 + 474*sqrt(5), so A274983(5) = 2*1060 = 2120 and a(5) = 2*474 = 948.

%t Round@Table[(QFactorial[n, GoldenRatio] - QFactorial[n, 1 - GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

%Y Cf. A274983, A005329, A275706, A276474, A276688, A276987.

%K nonn

%O 0,4

%A _Vladimir Reshetnikov_, Sep 23 2016