A058258 The 2-Up sequence: formed from final entries in rows of A058257.

1, 1, 1, 1, 3, 6, 26, 71, 413, 1456, 10576, 45541, 397023, 2020656, 20551376, 120686411, 1402815833, 9336345856, 122087570176, 908138776681, 13194844482843, 108480272749056, 1733786041150976, 15611712012050351, 272197308765744053, 2664103110372192256

Offset: 0

Author

N. J. A. Sloane, Dec 06 2000

Keywords

• nonn
• easy
• nice

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.

Crossrefs

Cf. A005981, A076417.

Column k=2 of A229892.

Programs

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 *)

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

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Dec 12 2000