A276755
a(n) = A275706(n)^2 + A276688(n)^2 = [n]{1+i}! * [n]{1-i}!, where [n]_q! is the q-factorial, i = sqrt(-1).
Original entry on oeis.org
1, 1, 5, 65, 1625, 66625, 4330625, 489360625, 110106140625, 52961053640625, 54285079981640625, 114704374001206640625, 484625980155098056640625, 4032572780870570929306640625
Offset: 0
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Table[QFactorial[n, 1 + I] QFactorial[n, 1 - I], {n, 0, 20}]
A275706
a(n) = Re([n]_{1+i}!), where [n]_q! is the q-factorial, i = sqrt(-1).
Original entry on oeis.org
1, 1, 2, 1, -40, 135, -860, 20145, -137100, -6726225, -212460900, -3642898575, 654642826500, -26505894416625, 3335048243533500, -1368325090374591375, 133951676745003682500, 123266968248328746879375, 63057521158814641016317500, 17732380504905960076345280625
Offset: 0
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a:= n-> Re(mul(((1+I)^j-1)/((1+I)-1), j=1..n)):
seq(a(n), n=0..20); # Alois P. Heinz, Sep 14 2016
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Re@Table[QFactorial[n, 1 + I], {n, 0, 20}]
A274983
a(n) = [n]phi! + [n]{1-phi}!, where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.
Original entry on oeis.org
2, 2, 3, 14, 130, 2120, 58120, 2636360, 196132320, 23805331920, 4698862837680, 1505416321070640, 781888977967152000, 657866357975539785600, 896265744457831561756800, 1976607903479486428467148800, 7055269158071576119808840371200
Offset: 0
For n = 3, [3]_phi! = 1060 + 474*sqrt(5), so a(5) = 2*1060 = 2120 and A274985(5) = 2*474 = 948.
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Round@Table[QFactorial[n, GoldenRatio] + QFactorial[n, 1 - GoldenRatio], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
A274985
a(n) = ([n]phi! - [n]{1-phi}!)/sqrt(5), where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.
Original entry on oeis.org
0, 0, 1, 6, 58, 948, 25992, 1179016, 87713040, 10646068080, 2101395344400, 673242645670320, 349671381118477440, 294206779308703578240, 400822226102433353285760, 883965927408694948620295680, 3155212287401150653204012531200
Offset: 0
For n = 3, [3]_phi! = 1060 + 474*sqrt(5), so A274983(5) = 2*1060 = 2120 and a(5) = 2*474 = 948.
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Round@Table[(QFactorial[n, GoldenRatio] - QFactorial[n, 1 - GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
Showing 1-4 of 4 results.