0,3

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).

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

Recursion: a(0) = 1; For n > 0

a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1).

a(n) = n!/floor(n/2)!^2

E.g.f.: (1+x)*BesselI(0, 2*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), 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(10) = 10!/5!^2 = trinomial(10,[5,0,5]);

a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).

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}.

Cf. A000142, A001405, A000188, A055772, A056042.

Bisections are A000984 and A002457.

Sequence in context: A070889 A072744 A056042 this_sequence A099566 A147299 A090549

Adjacent sequences: A056037 A056038 A056039 this_sequence A056041 A056042 A056043

nonn

Labos E. (labos(AT)ana.sote.hu), Jul 25 2000

Extended and edited by Peter Luschny (peter(AT)luschny.de), Jun 28 2009