A276991 - OEIS

%I #10 Feb 16 2025 08:33:36

%S 0,-1,0,-2,8,-86,1448,-40608,1867080,-140055600,17085644400,

%T -3383043446640,1085946439923840,-564694233102890880,

%U 475471874409018791040,-648068513405723438730240,1429638846930684965104992000,-5103811083889432701541321459200

%N a(n) = ((phi; phi)_n - (1-phi; 1-phi)_n)/sqrt(5), where (q; q)_n is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

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

%F (phi; phi)_n = (A276990(n) + a(n)*sqrt(5))/2.

%F (1-phi; 1-phi)_n = (A276990(n) - a(n)*sqrt(5))/2.

%F a(n) ~ c * (-1)^n * phi^(n*(n+1)/2) / sqrt(5), where c = (1/phi)_inf = A276987 = 0.1208019218617061294237231569887920563043992516794...

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

%Y Cf. A274983, A276474, A276987, A276990.

%K sign

%O 0,4

%A _Vladimir Reshetnikov_, Sep 24 2016