A058258 -id:A058258 - cp's OEIS frontend

A229892 Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 5, 3, 1, 1, 0, 16, 6, 4, 1, 1, 0, 61, 26, 10, 5, 1, 1, 0, 272, 71, 20, 15, 6, 1, 1, 0, 1385, 413, 125, 35, 21, 7, 1, 1, 0, 7936, 1456, 461, 70, 56, 28, 8, 1, 1, 0, 50521, 10576, 1301, 574, 126, 84, 36, 9, 1, 1

Offset: 0

Alois P. Heinz, Oct 02 2013

T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. T(n,k) = T(n,n) = A000012(n) = 1 for k>n.
T(2*n,n) = C(2*n-1,n) = A088218(n) = A001700(n-1) for n>0.
T(2*n+1,n) = C(2*n,n) = A000984(n).
T(2*n+1,n+1) = C(2n,n-1) = A001791(n) for n>0.

			Triangle T(n,k) begins:
  1;
  1,    1;
  0,    1,   1;
  0,    2,   1,   1;
  0,    5,   3,   1,  1;
  0,   16,   6,   4,  1,  1;
  0,   61,  26,  10,  5,  1, 1;
  0,  272,  71,  20, 15,  6, 1, 1;
  0, 1385, 413, 125, 35, 21, 7, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=1-10 give: A000111, A058258, A229884, A229885, A229886, A229887, A229888, A229889, A229890, A229891.

Cf. A227941, A229066, A229551.

b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=k,
       b(o-j, u+j-1, 1, k), b(u+j-1, o-j, t+1, k)), j=1..o))
    end:
T:= (n, k)-> `if`(k+1>=n, 1, `if`(k=0, 0, b(0, n, 0, k))):
seq(seq(T(n, k), k=0..n), n=0..10);

b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, Sum[If[t == k, b[o-j, u+j-1, 1, k], b[u+j-1, o-j, t+1, k]], {j, 1, o}]]; t[n_, k_] := If[k+1 >= n, 1, If[k == 0, 0, b[0, n, 0, k]]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

T(7,3) = 20: 1237654, 1247653, 1257643, 1267543, 1347652, 1357642, 1367542, 1457632, 1467532, 1567432, 2347651, 2357641, 2367541, 2457631, 2467531, 2567431, 3457621, 3467521, 3567421, 4567321.

A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.

Original entry on oeis.org

1, 1, 6, 71, 1456, 45541, 2020656, 120686411, 9336345856, 908138776681, 108480272749056, 15611712012050351, 2664103110372192256, 531909061958526321421, 122840808510269863827456, 32491881630252866646683891, 9758611490955498257378246656

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..100
Nicolas Basset, Counting and generating permutations using timed languages, 2013.
Nicolas Basset, Counting and generating permutations in regular classes of permutations, HAL Id: hal-01093994, 2014.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002; Moscow Math. J., 3 (2003), 647-659.
P. R. Stein & N. J. A. Sloane, Correspondence, 1975
Eric Weisstein's World of Mathematics, Generalized Hyperbolic Functions

Bisection of A058258.

Cf. A131453, A131454, A131455, A076417.

g:=((cosh(x)-1)*sin(x)+(cos(x)+1)*sinh(x))/(cos(x)*cosh(x)+1): series(%,x,35):
seq(n!*coeff(%,x,n),n=1..34,2); # Peter Luschny, Feb 07 2017

egf = ((Cosh[x]-1)*Sin[x]+(Cos[x]+1)*Sinh[x])/(Cos[x]*Cosh[x]+1); a[n_] := SeriesCoefficient[egf, {x, 0, 2*n+1}]*(2*n+1)!; Array[a, 17, 0] (* Jean-François Alcover, Mar 13 2014 *)

E.g.f.: x + Sum_{n>=1} a(n)*(x^(2n+1))/(2n+1)! = (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^(4k+j))/(4k+j)!, j = 0,1,2,3, is the j-th generalized hyperbolic function. - Peter Bala, Jul 13 2007
Basset (2013) gives an e.g.f. involving trigonometric and hyperbolic functions. - N. J. A. Sloane, Dec 24 2013
a(n) ~ 4 * (2*n+1)! / (tan(r/2)^2 * r^(2*n+2)), where r = A076417 = 1.8751040687119611664453082410782141625701117335310699882454137131... is the root of the equation cos(r)*cosh(r) = -1. - Vaclav Kotesovec, Feb 01 2015

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

Original entry on oeis.org

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

A058257 Triangle read by rows: this is a variant of A008280 in which 2 rows go from left to right, 2 from right to left, 2 from left to right, etc.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 3, 2, 1, 0, 0, 0, 3, 5, 6, 6, 6, 0, 0, 3, 8, 14, 20, 26, 71, 71, 71, 68, 60, 46, 26, 0, 413, 342, 271, 200, 132, 72, 26, 0, 0, 0, 413, 755, 1026, 1226, 1358, 1430, 1456, 1456, 1456, 0, 0, 413, 1168, 2194, 3420, 4778, 6208, 7664, 9120, 10576

Offset: 0

N. J. A. Sloane, Dec 06 2000

Suggested by Atkinson article in Information Processing Letters.

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  1, 1, 1, 0;
  3, 2, 1, 0, 0;
  0, 3, 5, 6, 6, 6;
  ...

M. D. Atkinson, Partial orders and comparison problems, Sixteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, (Boca Raton, Feb 1985), Congressus Numerantium 47, 77-88.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
M. D. Atkinson, Zigzag permutations and comparisons of adjacent elements, Information Processing Letters 21 (1985), 187-189.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (Abstract, pdf, ps).
Index entries for sequences related to boustrophedon transform

Cf. A058258, A008280, A000111.

a058257 n k = a058257_tabl !! n !! k
a058257_row n = a058257_tabl !! n
a058257_tabl = [1] : ox 0 [1] where
   ox turn xs = ys : ox (mod (turn + 1) 4) ys
      where ys | turn <= 1 = scanl (+) 0 xs
               | otherwise = reverse $ scanl (+) 0 $ reverse xs
-- Reinhard Zumkeller, Nov 01 2013

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

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A229892 Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.

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

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A058257 Triangle read by rows: this is a variant of A008280 in which 2 rows go from left to right, 2 from right to left, 2 from left to right, etc.

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