A274985 a(n) = ([n]phi! - [n]{1-phi}!)/sqrt(5), where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.

0, 0, 1, 6, 58, 948, 25992, 1179016, 87713040, 10646068080, 2101395344400, 673242645670320, 349671381118477440, 294206779308703578240, 400822226102433353285760, 883965927408694948620295680, 3155212287401150653204012531200

Offset: 0

Author

Vladimir Reshetnikov, Sep 23 2016

Keywords

• nonn

Examples

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

Links

• Eric Weisstein's World of Mathematics, q-Factorial, Golden Ratio.

Crossrefs

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

Programs

Mathematica

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

Formula

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

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

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