OFFSET
0,5
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.
FORMULA
E.g.f. (J. M. Luck, 2013): 1 + ((sin(x) - cos(x) + 1) * (cosh(x)-1) + (sin(x) + cos(x) + 1) * sinh(x)) / ((1 + cosh(x)*cos(x))). - Vaclav Kotesovec, Sep 06 2014
a(n) ~ c * n! / r^n, where r = A076417 = 1.8751040687119611664453... is the root of the equation cosh(r)*cos(r) = -1, and c = 4*cot(r/2)/r = 1.56598351207925... if n is even, c = 4*cot(r/2)^2/r = 1.14958147083780... if n is odd. - Vaclav Kotesovec, Sep 06 2014
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(`if`(t=2,
b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..30); # Alois P. Heinz, Oct 02 2013
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[If[t == 2, b[o-j, u+j-1, 1], b[u+j-1, o-j, t+1]], {j, 1, o}]] ; a[n_] := b[0, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)
CoefficientList[Series[1 + ((Sin[x]-Cos[x]+1) * (Cosh[x]-1) + (Sin[x]+Cos[x]+1) * Sinh[x]) / ((1+Cosh[x]*Cos[x])), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 06 2014 *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Dec 06 2000
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Dec 12 2000
STATUS
approved