A081123 a(n) = floor(n/2)!.
1, 1, 1, 1, 2, 2, 6, 6, 24, 24, 120, 120, 720, 720, 5040, 5040, 40320, 40320, 362880, 362880, 3628800, 3628800, 39916800, 39916800, 479001600, 479001600, 6227020800, 6227020800, 87178291200, 87178291200, 1307674368000, 1307674368000, 20922789888000, 20922789888000
Offset: 0
Examples
a(8) = 24, since 8 has 4 nondecreasing partitions with exactly two positive integer parts: (1,7),(2,6),(3,5),(4,4). Multiplying the first parts of these partitions together, we get: (1)(2)(3)(4) = 4! = 24. - _Wesley Ivan Hurt_, Jun 03 2013
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Magma
[Factorial(Floor(n/2)): n in [0..40]]; // Vincenzo Librandi, Aug 06 2013
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Maple
a:=n->floor(n/2)!: seq(a(k),k=1..70); # Wesley Ivan Hurt, Jun 03 2013
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Mathematica
Table[(Floor[n/2])!, {n, 0, 40}] (* Vincenzo Librandi, Aug 06 2013 *) -
PARI
for(n=0,50, print1((n\2)!, ", ")) \\ G. C. Greubel, Aug 01 2017
Formula
a(n) = floor(n/2)!.
E.g.f.: 1+sqrt(Pi)/2*(x+2)*exp(x^2/4)*erf(x/2). - Vladeta Jovovic, Sep 25 2003
From Sergei N. Gladkovskii, Jul 28 2012: (Start)
G.f. G(0) where G(k) = 1 + x/(1 - x*(k+1)/( x*(k+1) + 1/G(k+1))); (continued fraction, 3rd kind, 3-step ).
E.g.f. 1 + sqrt(Pi)/2*(x+2)*exp(x^2/4)*erf(x/2) = 1 + x/(G(0)-x) where G(k) = 2*k + 1 + x - (2*k+1)*x/(x + 2 - 2*x/G(k+1)); (continued fraction, 1st kind, 2-step).
(End)
G.f.: U(0) where U(k) = 1 + x/(1 - x*(k+2)/(x*(k+2) + 1/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 23 2012
G.f.: U(0) where U(k) = 1 + x/((2*k+1) - x*(2*k+1)/(x + 2*1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 23 2012
G.f.: 1 + x*G(0) where G(k) = 1 + x*(k+1)/(1 - x/(x + 1/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012
Comments