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))).

1, 1, 1, 1, 1, 4, 14, 34, 69, 496, 2896, 11056, 33661, 349504, 2856944, 14873104, 60376809, 819786496, 8615785216, 56814228736, 288294050521, 4835447317504, 62112775514624, 495812444583424, 3019098162602349, 60283564499562496, 915153344223809536, 8575634961418940416, 60921822444067346581, 1411083019275488149504, 24716980773496372066304

Offset: 0

Author

N. J. A. Sloane, Dec 31 2010

Keywords

• nonn

Comments

According to Mendes and Remmel, p. 56, this is the e.g.f. for 4-alternating permutations.

References

• 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/

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• J. M. Luck, On the frequencies of patterns of rises and falls, arXiv:1309.7764 [cond-mat.stat-mech], 2013-2014.
• Peter Luschny, An old operation on sequences: the Seidel transform

Crossrefs

Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), A181936 (m=5).

Cf. A181937.

Programs

Maple

A178964_list := proc(dim) local E,DIM,n,k;
DIM := dim-1; E := array(0..DIM, 0..DIM); E[0,0] := 1;
for n from 1 to DIM do
if n mod 4 = 0 then E[n,0] := 0 ;
   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;
else E[0,n] := 0;
   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;
fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..DIM)] end:
A178964_list(31); # Peter Luschny, Apr 02 2012
# Alternatively, using a bivariate exponential generating function:
A178964 := proc(n) local g, p, q;
g := (x,z) -> 2*exp(x*z)/(cosh(z)+cos(z));
p := (n,x) -> n!*coeff(series(g(x,z),z,n+2),z,n);
q := (n,m) -> if modp(n,m) = 0 then 0 else 1 fi:
(-1)^binomial(n,4)*p(n,q(n,4)) end:
seq(A178964(i),i=0..30); # Peter Luschny, Jun 06 2012

Mathematica

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 *)
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == 0,
     Sum[b[u - j, o + j - 1, Mod[t + 1, 4]], {j, 1, u}],
     Sum[b[u + j - 1, o - j, Mod[t + 1, 4]], {j, 1, o}]]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Apr 21 2021, after Alois P. Heinz in A250283 *)

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))))))

Sage

# Function A(m,n) defined in A181936.
A178964 = lambda n: (-1)^int(is_odd(n//4))*A(4,n)
print([A178964(n) for n in (0..30)]) # Peter Luschny, Jan 24 2017

Formula

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