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A056040
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Swinging factorial: 2^{n-(n mod 2)} prod_{1 <= k <= n} k^{(-1)^{k+1}}
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92
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1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 40116600, 1163381400, 155117520, 4808643120, 601080390
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of 'swinging orbitals' which are enumerated by the trinomial n over [floor(n/2), n mod 2, floor(n/2)].
Similar to but different from A001405(n) = binomial(n, floor(n/2)), a(n) = lcm(A001405(n-1), A001405(n)) (for n>0).
A055773(n) divides a(n), A001316(floor(n/2)) divides a(n).
sum(n>=0, 1/a(n)) = 4/3 + 8*Pi/(9*sqrt(3)). - Alexander R. Povolotsky, Aug 18 2012
Exactly p consecutive multiples of p follow the least positive multiple of p if p is an odd prime. Compare with the similar property of A100071. - Peter Luschny, Aug 27 2012
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REFERENCES
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Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..400
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FORMULA
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a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
a(n) = n!/floor(n/2)!^2
E.g.f.: (1+x)*BesselI(0, 2*x). - Vladeta Jovovic, Jan 19 2004
O.g.f.: a(n) = SeriesCoeff_{n}((1+z/(1-4*z^2))/sqrt(1-4*z^2)).
P.g.f.: a(n) = PolyCoeff_{n}((1+z^2)^n+n*z*(1+z^2)^(n-1)).
A056040(2n+1)=A046212(2n+1)=A100071(2n+1). - M. F. Hasler, Jan 25 2012
a(2*n) = binomial(2*n,n); a(2*n+1) = (2*n+1)*binomial(2*n,n). Central terms of triangle A211226. - Peter Bala, Apr 10 2012
Let f(n)=a(n-1), then f(n+4) = -(16*n*f(n))/(n+3)+(4*(n+2)*f(n+1))/(n+3)+(4*(2*n+3)*f(n+2))/(n+3)-((n+1)*f(n+3))/(n+3). - Alexander R. Povolotsky, Aug 17 2012
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+1)/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
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EXAMPLE
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a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).
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MAPLE
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SeriesCoeff := proc(s, n) series(s(w, n), w, n+2);
convert(%, polynom); coeff(%, w, n) end;
a1 := proc(n) local k;
2^(n-(n mod 2))*mul(k^((-1)^(k+1)), k=1..n) end:
a2 := proc(n) option remember;
`if`(n=0, 1, n^irem(n, 2)*(4/n)^irem(n+1, 2)*a2(n-1)) end;
a3 := n -> n!/iquo(n, 2)!^2;
g4 := z -> BesselI(0, 2*z)*(1+z);
a4 := n -> n!*SeriesCoeff(g4, n);
g5 := z -> (1+z/(1-4*z^2))/sqrt(1-4*z^2);
a5 := n -> SeriesCoeff(g5, n);
g6 := (z, n) -> (1+z^2)^n+n*z*(1+z^2)^(n-1);
a6 := n -> SeriesCoeff(g6, n);
a7 := n -> combinat[multinomial](n, floor(n/2), n mod 2, floor(n/2));
h := n -> binomial(n, floor(n/2)); # A001405
a8 := n -> ilcm(h(n-1), h(n));
seq(aN(i), i=0..32); # aN in {a1, a2, a3, a4, a5, a6, a7, a8}.
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MATHEMATICA
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f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* From Robert G. Wilson v, Aug 02 2010 *)
f[n_] := If[OddQ@n, n*Binomial[n - 1, (n - 1)/2], Binomial[n, n/2]]; Array[f, 33, 0] (* From Robert G. Wilson v, Aug 10 2010 *)
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PROG
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(PARI) a(n)=n!/(n\2)!^2; /* Charles R Greathouse IV, May 02, 2011 */
(MAGMA) [(Factorial(n)/(Factorial(Floor(n/2)))^2): n in [0..40]]; // Vincenzo Librandi, Sep 11 2011
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CROSSREFS
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Cf. A000142, A001405, A000188, A055772, A056042.
Bisections are A000984 and A002457. A211226.
Sequence in context: A070889 A072744 A056042 * A099566 A147299 A090549
Adjacent sequences: A056037 A056038 A056039 * A056041 A056042 A056043
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KEYWORD
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nonn,changed
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Jul 25 2000
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EXTENSIONS
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Extended and edited by Peter Luschny, Jun 28 2009
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STATUS
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approved
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