A005981 -id:A005981 - cp's OEIS frontend

A076417 Decimal expansion of first solution of equation cos(x)*cosh(x) = -1.

1, 8, 7, 5, 1, 0, 4, 0, 6, 8, 7, 1, 1, 9, 6, 1, 1, 6, 6, 4, 4, 5, 3, 0, 8, 2, 4, 1, 0, 7, 8, 2, 1, 4, 1, 6, 2, 5, 7, 0, 1, 1, 1, 7, 3, 3, 5, 3, 1, 0, 6, 9, 9, 8, 8, 2, 4, 5, 4, 1, 3, 7, 1, 3, 1, 0, 5, 6, 7, 9, 9, 5, 2, 8, 4, 0, 4, 2, 8, 6, 3, 8, 5, 2, 6, 5, 6, 6, 5, 5, 0, 5, 8, 1, 8, 8, 6, 0, 3, 7, 0, 8, 4, 1, 0

Offset: 1

Zak Seidov, Oct 10 2002

This is an equation related to a cantilever beam: cos(x)*cosh(x) = -1. The first three solutions are: 1.875 (this sequence), 4.69409 (A076418) and 7.854757 (A076419).

			1.87510406871196116644530824107821416257011173353...

Z. Guede, I. Elishakov, A fifth-order polynomial that serves as both buckling and vibration mode of an inhomogeneous structure, Chaos, Solitons and Fractals 12 (7) (2001) 1267-1298.

Cf. A009398, A076418, A076419.

Cf. A005981, A058258, A008775.

RealDigits[x/.FindRoot[Cos[x]Cosh[x]==-1,{x,1.8}, WorkingPrecision->120], 10,120][[1]] (* Harvey P. Dale, Jul 24 2011 *)

solve(x=1, 2, cos(x)*cosh(x) + 1) \\ Michel Marcus, Sep 11 2019

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

Original entry on oeis.org

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

N. J. A. Sloane, Dec 06 2000

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.

Cf. A005981, A076417.

Column k=2 of A229892.

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

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 *)

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

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

A131453 2 up, 2 down, ..., 2 up, 2 down permutations of length 4n+1.

Original entry on oeis.org

1, 6, 1456, 2020656, 9336345856, 108480272749056, 2664103110372192256, 122840808510269863827456, 9758611490955498257378246656, 1251231616578606273788469919481856, 245996119743058288132230759497577005056, 71155698830255977656506481145458378597728256

Offset: 0

Peter Bala, Jul 13 2007

Bisection of A005981.

			a(1) = 6. The six 2 up, 2 down permutations on 5 letters are (12543), (13542), (14532), (23541), (24532) and (34521).

Alois P. Heinz, Table of n, a(n) for n = 0..50
L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
H. Prodinger and T. A. Tshifhumulo, On q-Olivier Functions, Annals of Combinatorics 6 (2002), 181-194.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J. 3 (2003), 647-659.

Cf. A000111, A005981, A058257, A131454, A131455.

g:= (tan(x)+exp(2*x)*(tan(x)+1)-1)/(exp(2*x)+2*exp(x)*sec(x)+1): series(%,x,46):
seq(n!*coeff(%,x,n), n=1..45,4); # Peter Luschny, Feb 07 2017

Table[(CoefficientList[Series[((-1 + E^(2*x))*Cos[x] + (1 + E^(2*x))*Sin[x]) / (2*E^x + (1 + E^(2*x))* Cos[x]), {x, 0, 80}], x] * Range[0, 77]!)[[n]], {n, 2, 78, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)

E.g.f.: Sum_{n>=0} a(n)*(x^(4n+1))/(4n+1)! = (sin(x)*(exp(2x)+1)+cos(x)*(exp(2x)-1))/(2*exp(x)+cos(x)*(exp(2x)+1)).

A131454 2 up, 2 down, ..., 2 up, 2 down, 2 up permutations of length 4n+3.

Original entry on oeis.org

1, 71, 45541, 120686411, 908138776681, 15611712012050351, 531909061958526321421, 32491881630252866646683891, 3302814916156503291298772711761, 527393971346575736206847604137659031, 126355819963625435928020023737689391659701

Offset: 0

Peter Bala, Jul 13 2007

Bisection of A005981. The entries listed above suggest various congruences for a(n): a(n) = 1 (mod 10), a(n) = 1 + 70*n (mod 100), a(n) = 1 + 70*n + 200*n(n-1) (mod 1000). Are these congruences true for all n? For an arbitrary integer m, the sequence a(n) taken modulo m may eventually become periodic. Compare with A081727.

			(1 4 5 3 2 6 7) is a 2 up, 2 down, 2 up permutation - one of the seventy-one permutations of this type in the symmetric group on 7 letters.

Alois P. Heinz, Table of n, a(n) for n = 0..50
Christopher R. H. Hanusa, Alejandro H. Morales, and Martha Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
H. Prodinger and T. A. Tshifhumulo, On q-Olivier Functions, Annals of Combinatorics 6 (2002), 181-194.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J. 3 (2003), 647-659.

Cf. A000111, A005981, A058257, A131453, A131455.

g:=(sinh(x)-sin(x))/(cos(x)*cosh(x)+1): series(%,x,44):
seq(n!*coeff(%,x,n),n=3..45,4); # Peter Luschny, Feb 07 2017

Table[(CoefficientList[Series[(-Sin[x] + Sinh[x]) / (1 + Cos[x]*Cosh[x]), {x, 0, 60}], x] * Range[0, 59]!)[[n]], {n, 4, 60, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)

E.g.f.: Sum_{n>=0} a(n)*(x^(4n+3))/(4n+3)! = (exp(2x)-2*sin(x)*exp(x)-1)/(2*exp(x)+cos(x)*(exp(2x)+1)). It appears that a(n) = (4n+3)!*coefficient of x^(4n+3) in the Taylor expansion of -4/(2*exp(x)+cos(x)*(exp(2x)+1)).

A131455 Number of inequivalent properly oriented and labeled planar chord diagrams whose associated planar tree is a path on n + 1 vertices.

Original entry on oeis.org

1, 2, 18, 284, 7280, 273246, 14144592, 965491288, 84027112704, 9081387766810, 1193283000239616, 187340544144604212, 34633340434838499328, 7446726867419368499894, 1842612127654047957411840, 519870106084045866346942256, 165896395346243470375430193152, 59450668490817059243377908811698, 23773400714993519201980928470155264

Offset: 1

Peter Bala, Jul 13 2007

a(n) = n times the number of "2 up, 2 down" permutations of length 2*n-1 = n*A005981(n-1) for n >= 2.

a(n) ~ (c_1)*n*(2*n - 1)!/(c_2)^(2n), where c_1 is a constant and c_2 = 1.87510... is the smallest positive solution of the equation cos(z)* cosh(z) + 1 = 0.

			From _Petros Hadjicostas_, Jul 25 2020: (Start)
For n = 2, the a(2)/2 = 1 "2 up, 2 down" permutation of length 2*2 - 1 = 3 is the following:
         3
        /
       2
      /
     1
For n = 3, the a(3)/3 = 6 "2 up, 2 down" permutations of length 2*3 - 1 = 5 are the following:
        5          5          5          5          5          5
       / \        / \        / \        / \        / \        / \
      3   4      4   3      2   4      3   4      4   3      4   2
     /     \    /     \    /     \    /     \    /     \    /     \
    1       2  1       2  1       3  2       1  2       1  3       1
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..250
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J., 3 (2003), 647-659.
Eric Weisstein's World of Mathematics, Generalized Hyperbolic Functions.

Cf. A005981, A131453, A131454.

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-> n*b(0, 2*n-1, 0):
seq(a(n), n=1..19);  # Alois P. Heinz, Nov 23 2021

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_] := n*b[0, 2*n - 1, 0];
Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Mar 07 2022, after Alois P. Heinz *)

f(j,x,nn) = sum(k=0, 2*nn, (x^(4*k + j))/(4*k + j)!);
g(x,nn) = (x/2)*(f(0,x,nn)*f(1,x,nn) - f(2,x,nn)*f(3,x,nn) + f(3,x,nn))/(f(0,x,nn)^2 - f(1,x,nn)*f(3,x,nn));
lista(nn) = {default(seriesprecision, 2*nn); my(a=vector(nn)); for(n=1, nn, a[n] = (2*n)!*polcoef(Ser(g(x,nn)), 2*n)); a;} \\ Petros Hadjicostas, Jul 25 2020

E.g.f.: Sum_{n >= 1} a(n)*(x^(2*n))/(2*n)! = (x/2)*(f(0,x)*f(1,x) - f(2,x)*f(3,x) + f(3,x))/(f(0,x)^2 - f(1,x)*f(3,x)), where f(j,x) = Sum_{k >= 0} (x^(4*k + j))/(4*k + j)!, j = 0, 1, 2, 3, is the j-th generalized hyperbolic function.

More terms from Petros Hadjicostas, Jul 25 2020

A005983 Number of 4 up, 4 down, 4 up, ... permutations of length 4n+1.

Original entry on oeis.org

1, 1, 70, 26599, 33757360, 107709888805, 726401013530416, 9197888739246870571, 200656681438694771057920, 7065183006232334215872360169, 381446884048286939903298793116160, 30299510478473850351087119774475282895, 3422529682416045761005260546463028151218176

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, personal communication.

Alois P. Heinz, Table of n, a(n) for n = 0..132
P. R. Stein and N. J. A. Sloane, Correspondence, 1975

Cf. A229885.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(`if`(t=4,
      b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
    end:
a:= n-> b(0, 4*n+1, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 06 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[If[t == 4, b[o-j, u+j-1, 1], b[u+j-1, o-j, t+1]], {j, 1, o}]] ; a[n_] := b[0, 4*n+1, 0]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Nov 25 2014, after Alois P. Heinz *)

Typo in name fixed by Alois P. Heinz, Oct 06 2013

A005982 3 up, 3 down, 3 up, ... permutations of length 3n+1.

Original entry on oeis.org

1, 20, 1301, 202840, 61889101, 32676403052, 27418828825961, 34361404413755056, 61335081309931829401, 150221740688275657957940, 489799709605132718770274141, 2073641570051429601078643837960, 11163099186064084100687107863253381

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, personal communication.

Alois P. Heinz, Table of n, a(n) for n = 1..100
P. R. Stein & N. J. A. Sloane, Correspondence, 1975

Cf. A229884.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(`if`(t=3,
       b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
    end:
a:= n-> b(0, 3*n+1, 0):
seq(a(n), n=1..20);  # Alois P. Heinz, Oct 06 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[If[t == 3, b[o-j, u+j-1, 1], b[u+j-1, o-j, t+1]], {j, 1, o}]] ; a[n_] := b[0, 3*n+1, 0]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Nov 25 2014, after Alois P. Heinz *)

Typo in name fixed by Alois P. Heinz, Oct 06 2013

A259452 Number of 5 up, 5 down, 5 up, ... permutations of length 5n+1.

Original entry on oeis.org

1, 1, 252, 578005, 6190034016, 214265281290061, 19157603395806362772, 3800502511986185228829385, 1498722661993096106927612109936, 1081056808393919319749313795137642521, 1336319624105519211256870506149168604698792

Offset: 0

N. J. A. Sloane, Jun 28 2015

nonn

P. R. Stein, personal communication.

Alois P. Heinz, Table of n, a(n) for n = 0..100
P. R. Stein & N. J. A. Sloane, Correspondence, 1975

Cf. A005981-A005983.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(`if`(
       t=5, b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
    end:
a:= n-> b(0, 5*n+1, 0):
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 02 2015

k = 5; b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[If[t == k, b[o - j, u + j - 1, 1], b[u + j - 1, o - j, t + 1]], {j, 1, o}]]; Array[b[0, k # + 1, 0] &, 10] (* Michael De Vlieger, Oct 15 2017, after Jean-François Alcover at A005983 *)

More terms from Alois P. Heinz, Jul 02 2015

cp's OEIS Frontend

A076417 Decimal expansion of first solution of equation cos(x)*cosh(x) = -1.

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

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A131453 2 up, 2 down, ..., 2 up, 2 down permutations of length 4n+1.

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A131454 2 up, 2 down, ..., 2 up, 2 down, 2 up permutations of length 4n+3.

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A131455 Number of inequivalent properly oriented and labeled planar chord diagrams whose associated planar tree is a path on n + 1 vertices.

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A005983 Number of 4 up, 4 down, 4 up, ... permutations of length 4n+1.

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A005982 3 up, 3 down, 3 up, ... permutations of length 3n+1.

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