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A058258 The 2-Up sequence: formed from final entries in rows of A058257. 5
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 (list; graph; refs; listen; history; text; internal format)
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, 2013

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-Fran├žois 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 A214309 A005646

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

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Last modified November 20 23:06 EST 2019. Contains 329348 sequences. (Running on oeis4.)