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A178964 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))). 5

%I

%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, 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-Fran├žois Alcover_, Feb 25 2014 *)

%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

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Last modified April 19 20:04 EDT 2019. Contains 322291 sequences. (Running on oeis4.)