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

0,5

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

Cf. A005981, A076417.

Column k=2 of A229892.

Sequence in context: A148664 A148665 A148666 * A215397 A359974 A214309

Adjacent sequences: A058255 A058256 A058257 * A058259 A058260 A058261

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