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A056040
Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).
147
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, 19835652870, 2333606220
OFFSET
0,3
COMMENTS
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).
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
a(n) is the number of vertices of the polytope resulting from the intersection of an n-hypercube with the hyperplane perpendicular to and bisecting one of its long diagonals. - Didier Guillet, Jun 11 2018 [Edited by Peter Munn, Dec 06 2022]
LINKS
Peter Luschny, Orbitals.
Peter Luschny, Swinging Factorial.
FORMULA
a(n) = n!/floor(n/2)!^2. [Essentially the original name.]
a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
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)).
a(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
D-finite with recurrence: n*a(n) + (n-2)*a(n-1) + 4*(-2*n+3)*a(n-2) + 4*(-n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - Alexander R. Povolotsky, Aug 17 2012
Sum_{n>=0} 1/a(n) = 4/3 + 8*Pi/(9*sqrt(3)). - Alexander R. Povolotsky, Aug 18 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
Central column of the coefficients of the swinging polynomials A162246. - Peter Luschny, Oct 22 2013
a(n) = Sum_{k=0..n} A189231(n, 2*k). (Cf. A212303 for the odd case.) - Peter Luschny, Oct 30 2013
a(n) = hypergeometric([-n,-n-1,1/2],[-n-2,1],2)*2^(n-1)*(n+2). - Peter Luschny, Sep 22 2014
a(n) = 4^floor(n/2)*hypergeometric([-floor(n/2), (-1)^n/2], [1], 1). - Peter Luschny, May 19 2015
Sum_{n>=0} (-1)^n/a(n) = 4/3 - 4*Pi/(9*sqrt(3)). - Amiram Eldar, Mar 10 2022
EXAMPLE
a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).
MAPLE
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));
F := [a1, a2, a3, a4, a5, a6, a7, a8];
for a in F do seq(a(i), i=0..32) od;
MATHEMATICA
f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* 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] (* Robert G. Wilson v, Aug 10 2010 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; (* or, twice faster: *) sf[n_] := n!/Quotient[n, 2]!^2; Table[sf[n], {n, 0, 32}] (* Jean-François Alcover, Jul 26 2013, updated Feb 11 2015 *)
PROG
(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
(SageMath)
def A056040():
r, n = 1, 0
while True:
yield r
n += 1
r *= 4/n if is_even(n) else n
a = A056040(); [next(a) for i in range(36)] # Peter Luschny, Oct 24 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Jul 25 2000
EXTENSIONS
Extended and edited by Peter Luschny, Jun 28 2009
STATUS
approved