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A123029
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a(2*n-1) = prod(i=1..n, fibonacci(2*i-1)) and a(2*n) = prod(i=1..n, fibonacci(2*i)).
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4
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1, 1, 2, 3, 10, 24, 130, 504, 4420, 27720, 393380, 3991680, 91657540, 1504863360, 55911099400, 1485300136320, 89290025741800, 3838015552250880, 373321597626465800, 25964175210977203200, 4086378207619294646800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| From Johannes W. Meijer, Aug 21 2011: (Start)
An appropriate name for this sequence is Fibonacci double factorial, cf. A006882.
In Park’s article "A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov" in appendix 2 a number triangle T(n,k) with T(n,n) = a(n+1), n>=0, appears if we assume that b(r) = fibonacci(r); see A103631 and A194005. (End)
The original name of this sequence was: A000045 inside a second linear differential equation recursion: b(n) = b(n-1) + b(n-2) --> Binet(n) of A000045 a(n) = b(n)*a(n-2)/(n*(n-1)).
Bagula also stated that using the solutions to these second order differential equations Markov/ linear recursions can be encoded as analog functions.
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LINKS
| P.C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702.
Eric Weisstein, Fibonorial Mathworld.
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FORMULA
| a(n) = n!*a1(n) with a1(n) = b(n)*a1(n-2)/(n*(n-1)), a1(0) = 1, a1(1) = 1; b(n) = b(n-1) + b(n-2), b(0) = 0, b(1) = 1 and b(n) = F(n) with F(n) = A000045(n).
From Johannes W. Meijer, Aug 21 2011: (Start)
a(2*n-1) = product(F(2*i-1), i =1..n) and a(2*n) = product(F(2*i), i=1..n)
a(n) = F(n)*a(n-2)
a(2*n) = A194157(n) and a(2*n-1) = A194158(n) (End)
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MAPLE
| with(combinat): A123029 :=proc(n): if type(n, even) then mul(fibonacci(2*i), i=1..n/2) else mul(fibonacci(2*i-1), i= 1..(n+1)/2) fi: end: seq(A123029(n), n=1..21); # [Johannes W. Meijer, Aug 21 2011]
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MATHEMATICA
| f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == a[n - 1] + a[n - 2], a[0] == 0, a[1] == 1}, a[n], n][[1]] // FullSimplify] ; Clear[a] a[n_] := a[n] = f[n]*a[n - 2]/(n*(n - 1)); a[0] = 1; a[1] = 1; Table[ExpandAll[a[n]*n! ], {n, 0, 30}]
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CROSSREFS
| Cf. A000045.
Cf. A003266, A194157 and A194158 [Johannes W. Meijer, Aug 21 2011]
Sequence in context: A162034 A105286 A059929 * A103018 A005158 A182926
Adjacent sequences: A123026 A123027 A123028 * A123030 A123031 A123032
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KEYWORD
| nonn,easy
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AUTHOR
| Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 25 2006
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EXTENSIONS
| Edited and information added by Johannes W. Meijer, Aug 21 2011
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