A276991 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.

0, -1, 0, -2, 8, -86, 1448, -40608, 1867080, -140055600, 17085644400, -3383043446640, 1085946439923840, -564694233102890880, 475471874409018791040, -648068513405723438730240, 1429638846930684965104992000, -5103811083889432701541321459200

Offset: 0

Author

Vladimir Reshetnikov, Sep 24 2016

Keywords

• sign

Links

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

Crossrefs

Cf. A274983, A276474, A276987, A276990.

Programs

Mathematica

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 *)

Formula

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

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

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