|
%I #44 Apr 21 2021 13:17:38
%S 1,1,1,1,1,4,14,34,69,496,2896,11056,33661,349504,2856944,14873104,
%T 60376809,819786496,8615785216,56814228736,288294050521,4835447317504,
%U 62112775514624,495812444583424,3019098162602349,60283564499562496,915153344223809536,8575634961418940416,60921822444067346581,1411083019275488149504,24716980773496372066304
%N E.g.f.: (1+sqrt(2)*sin(x/sqrt(2))*cosh(x/sqrt(2))+sin(x/sqrt(2))*sinh(x/sqrt(2)))/(cos(x/sqrt(2))*cosh(x/sqrt(2))).
%C According to Mendes and Remmel, p. 56, this is the e.g.f. for 4-alternating permutations.
%D Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page http://math.ucsd.edu/~remmel/
%H Alois P. Heinz, <a href="/A178964/b178964.txt">Table of n, a(n) for n = 0..200</a>
%H J. M. Luck, <a href="http://arxiv.org/abs/1309.7764">On the frequencies of patterns of rises and falls</a>, arXiv:1309.7764 [cond-mat.stat-mech], 2013-2014.
%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) ~ n! * 2^(n/2+1) * (-sqrt(2)*(-1+(-1)^n) - 2*cos(n*Pi/2)*(sinh(Pi/2)-1)/cosh(Pi/2) + (1+(-1)^n)*(1 + sinh(Pi/2))/cosh(Pi/2)) / Pi^(n+1). - _Vaclav Kotesovec_, Sep 09 2014
%p A178964_list := proc(dim) local E,DIM,n,k;
%p DIM := dim-1; E := array(0..DIM, 0..DIM); E[0,0] := 1;
%p for n from 1 to DIM do
%p if n mod 4 = 0 then E[n,0] := 0 ;
%p for k from n-1 by -1 to 0 do E[k,n-k] := E[k+1,n-k-1] + E[k,n-k-1] od;
%p else E[0,n] := 0;
%p for k from 1 by 1 to n do E[k,n-k] := E[k-1,n-k+1] + E[k-1,n-k] od;
%p fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..DIM)] end:
%p A178964_list(31); # _Peter Luschny_, Apr 02 2012
%p # Alternatively, using a bivariate exponential generating function:
%p A178964 := proc(n) local g, p, q;
%p g := (x,z) -> 2*exp(x*z)/(cosh(z)+cos(z));
%p p := (n,x) -> n!*coeff(series(g(x,z),z,n+2),z,n);
%p q := (n,m) -> if modp(n,m) = 0 then 0 else 1 fi:
%p (-1)^binomial(n,4)*p(n,q(n,4)) end:
%p seq(A178964(i),i=0..30); # _Peter Luschny_, Jun 06 2012
%t max = 30; s = Series[Sec[x]*Sech[x]+Tan[x]*(Sqrt[2]+Tanh[x]) /. x -> x/Sqrt[2], {x, 0, max+1}]; a[n_] := SeriesCoefficient[s, {x, 0, n}]*n!; Table[a[n], {n, 0, max}] (* _Jean-François Alcover_, Feb 25 2014 *)
%t b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == 0,
%t Sum[b[u - j, o + j - 1, Mod[t + 1, 4]], {j, 1, u}],
%t Sum[b[u + j - 1, o - j, Mod[t + 1, 4]], {j, 1, o}]]];
%t a[n_] := b[n, 0, 0];
%t a /@ Range[0, 35] (* _Jean-François Alcover_, Apr 21 2021, after _Alois P. Heinz_ in A250283 *)
%o (Sage)
%o # Function A(m,n) defined in A181936.
%o A178964 = lambda n: (-1)^int(is_odd(n//4))*A(4,n)
%o print([A178964(n) for n in (0..30)]) # _Peter Luschny_, Jan 24 2017
%o (PARI) x='x+O('x^30);round(Vec(serlaplace((1+sqrt(2)*sin(x/sqrt(2))*cosh( x/sqrt(2)) + sin(x/sqrt(2))*sinh(x/sqrt(2)))/(cos(x/sqrt(2))*cosh(x/sqrt(2))))))
%Y Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), A181936 (m=5).
%Y Cf. A181937.
%K nonn
%O 0,6
%A _N. J. A. Sloane_, Dec 31 2010
|
