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n-alternating permutations of length 3n.
4

%I #12 Aug 13 2015 14:50:29

%S 1,61,1513,33661,750751,17116009,398840401,9464040829,227864057851,

%T 5550936701311,136526608389601,3384729259165801,84478081828015513,

%U 2120572560190269841,53494979095639780513,1355345459896317255037,34469858667289041256051,879619727291950363099291

%N n-alternating permutations of length 3n.

%C a(n) = A181985(n,3).

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>

%F a(n) = (3*n)!*(1/(3*n)!-2/(n!*(2*n)!)+1/(n!)^3). - _Peter Luschny_, Aug 13 2015

%p A211213 := proc(n) local E, dim, i, k; dim := 3*n;

%p E := array(0..dim, 0..dim); E[0, 0] := 1;

%p for i from 1 to dim do

%p if i mod n = 0 then E[i, 0] := 0 ;

%p for k from i-1 by -1 to 0 do E[k, i-k] := E[k+1, i-k-1] + E[k, i-k-1] od;

%p else E[0, i] := 0;

%p for k from 1 by 1 to i do E[k, i-k] := E[k-1, i-k+1] + E[k-1, i-k] od;

%p fi od; E[0, dim] end:

%p seq(A211213(n), n = 1..18);

%p # Alternatively:

%p a := x -> (3*x)!*(1/(3*x)!-2/(x!*(2*x)!)+1/(x!)^3):

%p seq(a(n),n=1..18); # _Peter Luschny_, Aug 13 2015

%t nmax = 18; a[n_] := Module[{e, dim = n*(nmax-1)}, e[0, 0] = 1; For[i = 1, i <= dim, i++, If[Mod[i, n] == 0 , e[i, 0] = 0; For[k = i-1, k >= 0, k--, e[k, i-k] = e[k+1, i-k-1] + e[k, i-k-1] ], e[0, i] = 0; For[k = 1, k <= i, k++, e[k, i-k] = e[k-1, i-k+1] + e[k-1, i-k] ] ]]; e[0, 3*n]] ; Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Jul 26 2013, after Maple *)

%Y Cf. A181985, A030662, A181991.

%K nonn

%O 1,2

%A _Peter Luschny_, Apr 05 2012