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A056040 Swinging factorial, a(n) = 2^(n-(n mod 2))*prod_{1 <= k <= n} k^((-1)^(k+1)). 129
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 (list; graph; refs; listen; history; text; internal format)
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).

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

REFERENCES

Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Peter Luschny, Swinging Factorial

FORMULA

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)).

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

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

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

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-Fran├žois 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

(Sage)

def A056040():

    r, n = 1, 0

    while True:

        yield r

        n += 1

        r *= 4/n if is_even(n) else n

a = A056040(); [a.next() for i in range(36)]  # Peter Luschny, Oct 24 2013

CROSSREFS

Bisections are A000984 and A002457.

Cf. A000142, A001405, A000188, A055772, A056042, A211226, A162246, A189231, A212303.

Sequence in context: A070889 A072744 A056042 * A099566 A147299 A090549

Adjacent sequences:  A056037 A056038 A056039 * A056041 A056042 A056043

KEYWORD

nonn

AUTHOR

Labos Elemer, Jul 25 2000

EXTENSIONS

Extended and edited by Peter Luschny, Jun 28 2009

STATUS

approved

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Last modified March 24 00:18 EDT 2017. Contains 283983 sequences.