A178963 -id:A178963 - cp's OEIS frontend

A181937 André numbers. Square array A(n,k), n>=2, k>=0, read by antidiagonals upwards, A(n,k) = n-alternating permutations of length k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 16, 1, 1, 1, 1, 1, 9, 61, 1, 1, 1, 1, 1, 4, 19, 272, 1, 1, 1, 1, 1, 1, 14, 99, 1385, 1, 1, 1, 1, 1, 1, 5, 34, 477, 7936, 1, 1, 1, 1, 1, 1, 1, 20, 69, 1513, 50521, 1, 1, 1, 1, 1, 1, 1, 6, 55, 496, 11259, 353792

Offset: 0

Peter Luschny, Apr 03 2012

The André numbers were studied by Désiré André in the case n=2 around 1880. The present author suggests that the numbers A(n,k) be named in honor of André. Already in 1877 Ludwig Seidel gave an efficient algorithm for computing the coefficients of secant and tangent which immediately carries over to the general case. Anthony Mendes and Jeffrey Remmel give exponential generating functions for the general case.

			n\k [0][1][2][3][4] [5] [6]  [7]   [8]   [9]  [10]    [11]
[1]  1, 1, 1, 1, 1,  1,  1,   1,    1,    1,    1,       1  [A000012]
[2]  1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792  [A000111]
[3]  1, 1, 1, 1, 3,  9, 19,  99,  477, 1513, 11259,  74601  [A178963]
[4]  1, 1, 1, 1, 1,  4, 14,  34,   69,  496,  2896,  11056  [A178964]
[5]  1, 1, 1, 1, 1,  1,  5,  20,   55,  125,   251,   2300  [A181936]
[6]  1, 1, 1, 1, 1,  1,  1,   6,   27,   83,   209,    461  [A250283]

Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page.

Alois P. Heinz, Antidiagonals k = 0..140, flattened
Désiré André, Développement de séc x et de tang x, C. R. Math. Acad. Sci. Paris 88 (1879), 965-967.
Désiré André, Sur les permutations alternées, J. Math. pur. appl., 7 (1881), 167-184.
Peter Luschny, An old operation on sequences: the Seidel transform.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]

Cf. A000111, A178963, A178964, A181936, A250283, A250284, A250285, A250286, A250287.

# Signed version.
using Memoize
@memoize function André(m, n)
    n ≤ 0 && return 1
    r = range(0, stop=n-1, step=m)
    S = sum(binomial(n, k) * André(m, k) for k in r)
    n % m == 0 ? -S : S
end
for m in 1:8 println([André(m, n) for n in 0:11]) end # Peter Luschny, Feb 09 2019

A181937_list := proc(n, len) local E,dim,i,k;  # Seidel's boustrophedon transform
dim := len-1; E := array(0..dim, 0..dim); E[0,0] := 1;
for i from 1 to dim do
if i mod n = 0 then E[i,0] := 0 ;
   for k from i-1 by -1 to 0 do E[k,i-k] := E[k+1,i-k-1] + E[k,i-k-1] od;
else E[0,i] := 0;
   for k from 1 by 1 to i do E[k,i-k] := E[k-1,i-k+1] + E[k-1,i-k] od;
fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..dim)] end:
for n from 2 to 6 do print(A181937_list(n,12)) od;
# Alternative, with an additional row 0:
Andre := proc(m, n) option remember; local k; ifelse(n <= 0, 1, ifelse(m = 0, 1,
-add(binomial(n, k) * Andre(m, k), k = 0..n-1, m))) end:
T := (n, k) -> abs(Andre(n, k)): seq(lprint(seq(T(n, k), k = 0..11)), n = 0..9);
# Peter Luschny, Aug 19 2024

dim = 13; e[][0, 0] = 1; e[m][n_ /; 0 <= n <= dim, 0] /; Mod[n, m] == 0 = 0; e[m_][k_ /; 0 <= k <= dim, n_ /; 0 <= n <= dim] /; Mod[n+k, m] == 0 := e[m][k, n] = e[m][k, n-1] + e[m][k+1, n-1]; e[m_][0, n_ /; 0 <= n <= dim] /; Mod[n, m] == 0 = 0; e[m_][k_ /; 0 <= k <= dim, n_ /; 0 <= n <= dim] /; Mod[n+k, m] != 0 := e[m][k, n] = e[m][k-1, n] + e[m][k-1, n+1]; e[][, ] = 0; a[, 0] = 1; a[m_, n_] := e[m][n, 0] + e[m][0, n]; Table[a[m-n+1, n], {m, 1, dim-1}, {n, 0, m-1}] // Flatten (* Jean-François Alcover, Jul 23 2013, after Maple *)
b[r_, u_, o_, t_] := b[r, u, o, t] = If[u + o == 0, 1, If[t == 0, Sum[b[r, u - j, o + j - 1, Mod[t + 1, r]], {j, 1, u}], Sum[b[r, u + j - 1, o - j, Mod[t + 1, r]], {j, 1, o}]]]; A[n_, k_] := b[n, k, 0, 0];
Table[A[n - k, k], {n, 2, 13}, {k, 0, n - 2}] // Flatten
(* Jean-François Alcover, Nov 22 2023, after Alois P. Heinz in A250283 *)
Andre[n_, k_] := Andre[n, k] = If[k <= 0, 1, If[n == 0, 1, -Sum[Binomial[k, j] Andre[n, j], {j, 0, k-1, n}]]];
Table[Abs[Andre[n, k]], {n, 0, 6}, {k, 0, 11}] // MatrixForm
(* Peter Luschny, Aug 19 2024 *)

@cached_function
def A(m, n):
    if n == 0: return 1
    s = -1 if m.divides(n) else 1
    t = [m*k for k in (0..(n-1)//m)]
    return s*add(binomial(n, k)*A(m, k) for k in t)
A181937_row = lambda m, n: (-1)^int(is_odd(n//m))*A(m, n)
for n in (1..6): print([A181937_row(n, k) for k in (0..20)]) # Peter Luschny, Feb 06 2017

A002115 Generalized Euler numbers.

Original entry on oeis.org

1, 1, 19, 1513, 315523, 136085041, 105261234643, 132705221399353, 254604707462013571, 705927677520644167681, 2716778010767155313771539, 14050650308943101316593590153, 95096065132610734223282520762883, 823813936407337360148622860507620561

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..166
Takao Komatsu and Ram Krishna Pandey, On hypergeometric Cauchy numbers of higher grade, AIMS Mathematics (2021) Vol. 6, Issue 7, 6630-6646.
Takao Komatsu and Guo-Dong Liu, Congruence properties of Lehmer-Euler numbers, arXiv:2501.01178 [math.NT], 2025.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
Bruce E. Sagan, Generalized Euler numbers and ordered set partitions, arXiv:2501.07692 [math.NT], 2025. See p. 3.

Cf. A000364, A178963, A278073.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, `if`(t=0,
       add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
       add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
    end:
a:= n-> b(3*n, 0$2):
seq(a(n), n=0..17);  # Alois P. Heinz, Aug 12 2019
# Alternative:
h := 1 / hypergeom([], [1/3, 2/3], (-x/3)^3): ser := series(h, x, 40):
seq((3*n)! * coeff(ser, x, 3*n), n = 0..13); # Peter Luschny, Mar 13 2023

max = 12; f[x_] := 1/(1/3*Exp[-x^(1/3)] + 2/3*Exp[1/2*x^(1/3)]*Cos[1/2*3^(1/2)* x^(1/3)]); CoefficientList[Series[f[x], {x, 0, max}], x]*(3 Range[0, max])! (* Jean-François Alcover, Sep 16 2013, after Vladeta Jovovic *)

E.g.f.: Sum_{n >= 0} a(n)*x^n/(3*n)! = 1/((1/3)*exp(-x^(1/3)) + (2/3)*exp((1/2)*x^(1/3))*cos((1/2)*3^(1/2)*x^(1/3))). - Vladeta Jovovic, Feb 13 2005
E.g.f.: 1/U(0) where U(k) = 1 - x/(6*(6*k+1)*(3*k+1)*(2*k+1) - 6*x*(6*k+1)*(3*k+1)*(2*k+1)/(x - 12*(6*k+5)*(3*k+2)*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2012
Alternating row sums of A278073. - Peter Luschny, Sep 07 2017
a(n) = A178963(3n). - Alois P. Heinz, Aug 12 2019
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(3*n,3*k) * a(n-k). - Ilya Gutkovskiy, Jan 27 2020
a(n) = (3*n)! * [x^(3*n)] hypergeom([], [1/3, 2/3], (-x/3)^3)^(-1). - Peter Luschny, Mar 13 2023

More terms from Vladeta Jovovic, Feb 13 2005

A215064 Triangle read by rows, e.g.f. exp(xz)((exp(x/2)+exp(x3/2))/((exp(3x/2)+ 2cos(sqrt(3)x/2))/3)-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, -1, 3, 3, 1, -3, -4, 6, 4, 1, -9, -15, -10, 10, 5, 1, 19, -54, -45, -20, 15, 6, 1, 99, 133, -189, -105, -35, 21, 7, 1, 477, 792, 532, -504, -210, -56, 28, 8, 1, -1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1, -11259

Offset: 0

Peter Luschny, Aug 01 2012

			[0] [1]
[1] [1, 1]
[2] [1, 2, 1]
[3] [-1, 3, 3, 1]
[4] [-3, -4, 6, 4, 1]
[5] [-9, -15, -10, 10, 5, 1]
[6] [19, -54, -45, -20, 15, 6, 1]
[7] [99, 133, -189, -105, -35, 21, 7, 1]
[8] [477, 792, 532, -504, -210, -56, 28, 8, 1]
[9] [-1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1]

Cf. A215060, A215061, A215062, A215063, A215065.

max = 11; f = Exp[x*z]*((Exp[x/2] + Exp[x*(3/2)])/((Exp[3*(x/2)] + 2*Cos[Sqrt[3]*(x/2)])/3) - 1); coes = CoefficientList[ Series[f, {x, 0, max}, {z, 0, max}], {x, z}]; Table[ coes[[n, k]]*(n - 1)!, {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

# uses[triangle from A215060]
def A215064_triangle(dim):
    var('x, z')
    f = exp(x*z)*((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)-1)
    return triangle(f, dim)
A215064_triangle(12)

Matrix inverse is A215065.
T(n,k) = A215060(n,k) + A215062(n,k) - [n==k].
|T(n,0)| = A178963(n).
|T(3*n,0)| = A002115(n).

A249402 The number of 3-alternating permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 3, 11, 40, 99, 589, 3194, 11259, 92159, 666160, 3052323, 31799041, 287316122, 1620265923, 20497038755, 222237912664, 1488257158851, 22149498351205, 280180369563194, 2172534146099019, 37183508549366519, 537546603651987424, 4736552519729393091

Offset: 0

R. J. Mathar, Oct 27 2014

nonn

A sequence a(1),a(2),... is called k-alternating if a(i) > a(i+1) iff i=1 (mod k). a(n) gives the number of 3-alternating permutations of [n].

			The a(4)=3 3-alternating permutations of [4] are: [2 1 3 4 ] [3 1 2 4 ] and [4 1 2 3 ].
The a(5)=11 3-alternating permutations of [5] are: [2 1 3 5 4 ] [2 1 4 5 3 ] [3 1 2 5 4 ] [3 1 4 5 2 ] [3 2 4 5 1 ] [4 1 2 5 3 ] [4 1 3 5 2 ] [4 2 3 5 1 ] [5 1 2 4 3 ] [5 1 3 4 2 ] and [5 2 3 4 1 ].

Alois P. Heinz, Table of n, a(n) for n = 0..500
R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240, page 17.

Cf. A065619 (2-alternating).
Cf. A178963, A249583 (alternative definitions of 3-alternating permutations).
Column k=3 of A250261.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=1, add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 27 2014

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, If[t == 1, Sum[b[u-j, o+j-1, Mod[t+1, 3]], {j, 1, u}], Sum[b[u+j-1, o-j, Mod[t+1, 3]], {j, 1, o}]]]; a[n_] := b[0, n, 0]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

a(16)-a(25) from Alois P. Heinz, Oct 27 2014

A178964 E.g.f.: (1+sqrt(2)sin(x/sqrt(2))cosh(x/sqrt(2))+sin(x/sqrt(2))sinh(x/sqrt(2)))/(cos(x/sqrt(2))cosh(x/sqrt(2))).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 14, 34, 69, 496, 2896, 11056, 33661, 349504, 2856944, 14873104, 60376809, 819786496, 8615785216, 56814228736, 288294050521, 4835447317504, 62112775514624, 495812444583424, 3019098162602349, 60283564499562496, 915153344223809536, 8575634961418940416, 60921822444067346581, 1411083019275488149504, 24716980773496372066304

Offset: 0

N. J. A. Sloane, Dec 31 2010

nonn

According to Mendes and Remmel, p. 56, this is the e.g.f. for 4-alternating permutations.

Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page http://math.ucsd.edu/~remmel/

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.
Peter Luschny, An old operation on sequences: the Seidel transform

Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), A181936 (m=5).

Cf. A181937.

A178964_list := proc(dim) local E,DIM,n,k;
DIM := dim-1; E := array(0..DIM, 0..DIM); E[0,0] := 1;
for n from 1 to DIM do
if n mod 4 = 0 then E[n,0] := 0 ;
   for k from n-1 by -1 to 0 do E[k,n-k] := E[k+1,n-k-1] + E[k,n-k-1] od;
else E[0,n] := 0;
   for k from 1 by 1 to n do E[k,n-k] := E[k-1,n-k+1] + E[k-1,n-k] od;
fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..DIM)] end:
A178964_list(31); # Peter Luschny, Apr 02 2012
# Alternatively, using a bivariate exponential generating function:
A178964 := proc(n) local g, p, q;
g := (x,z) -> 2*exp(x*z)/(cosh(z)+cos(z));
p := (n,x) -> n!*coeff(series(g(x,z),z,n+2),z,n);
q := (n,m) -> if modp(n,m) = 0 then 0 else 1 fi:
(-1)^binomial(n,4)*p(n,q(n,4)) end:
seq(A178964(i),i=0..30); # Peter Luschny, Jun 06 2012

max = 30; s = Series[Sec[x]*Sech[x]+Tan[x]*(Sqrt[2]+Tanh[x]) /. x -> x/Sqrt[2], {x, 0, max+1}]; a[n_] := SeriesCoefficient[s, {x, 0, n}]*n!; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 25 2014 *)
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == 0,
     Sum[b[u - j, o + j - 1, Mod[t + 1, 4]], {j, 1, u}],
     Sum[b[u + j - 1, o - j, Mod[t + 1, 4]], {j, 1, o}]]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Apr 21 2021, after Alois P. Heinz in A250283 *)

x='x+O('x^30);round(Vec(serlaplace((1+sqrt(2)*sin(x/sqrt(2))*cosh( x/sqrt(2)) + sin(x/sqrt(2))*sinh(x/sqrt(2)))/(cos(x/sqrt(2))*cosh(x/sqrt(2))))))

# Function A(m,n) defined in A181936.
A178964 = lambda n: (-1)^int(is_odd(n//4))*A(4,n)
print([A178964(n) for n in (0..30)]) # Peter Luschny, Jan 24 2017

a(n) ~ n! * 2^(n/2+1) * (-sqrt(2)*(-1+(-1)^n) - 2*cos(n*Pi/2)*(sinh(Pi/2)-1)/cosh(Pi/2) + (1+(-1)^n)*(1 + sinh(Pi/2))/cosh(Pi/2)) / Pi^(n+1). - Vaclav Kotesovec, Sep 09 2014

A181936 Number of 5-alternating permutations.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 5, 20, 55, 125, 251, 2300, 15775, 70500, 249250, 750751, 10006375, 97226875, 601638125, 2886735625, 11593285251, 202808749375, 2550175096250, 20163891580625, 122209131374375, 613498040952501, 13287626090593750, 205055676105734375

Offset: 0

Peter Luschny, Apr 03 2012

nonn

For an integer n>0, a permutation s = s_1...s_k is a n-alternating permutation if it has the property that s_i < s_{i+1} if and only if n divides i.

Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page.

Alois P. Heinz, Table of n, a(n) for n = 0..200
R. J. Cano, PARI Sequencer program.
Peter Luschny, An old operation on sequences: the Seidel transform.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]

Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), this sequence (m=5), A250283 (m=6), A250284 (m=7), A250285 (m=8), A250286 (m=9), A250287 (m=10).

Row n=5 of A181937.

A181936_list := proc(dim) local E,DIM,n,k;
DIM := dim-1; E := array(0..DIM, 0..DIM); E[0,0] := 1;
for n from 1 to DIM do
if n mod 5 = 0 then E[n,0] := 0 ;
   for k from n-1 by -1 to 0 do E[k,n-k] := E[k+1,n-k-1] + E[k,n-k-1] od;
else E[0,n] := 0;
   for k from 1 by 1 to n do E[k,n-k] := E[k-1,n-k+1] + E[k-1,n-k] od;
fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..DIM)] end:
A181936_list(28);
# Alternatively, using an exponential generating function:
A181936_list := proc(n) local H,F,i; H := (r,s) -> hypergeom(r,s/5,-(t/5)^5);
F := t -> 1+(t^5*H([1],[6,7,8,9,10])+5*t^4*H([],[6,7,8,9])+20*t^3*H([],[4,6,7,8])+60*t^2*H([],[3,4,6,7])+120*t^1*H([],[2,3,4,6]))/(120*H([],[2,3,4,1])); seq(i!*coeff(series(F(t),t,n+1),t,i),i=0..n-1) end:

dim = 27; e[0, 0] = 1; e[n_ /; Mod[n, 5] == 0 && 0 <= n <= dim, 0] = 0; e[k_ /; 0 <= k <= dim, n_ /; 0 <= n <= dim] /; Mod[n+k, 5] == 0 := e[k, n] = e[k, n-1] + e[k+1, n-1]; e[0, n_ /; Mod[n, 5] == 0 && 0 <= n <= dim] = 0; e[k_ /; 0 <= k <= dim, n_ /; 0 <= n <= dim] /; Mod[n+k, 5] != 0 := e[k, n] = e[k-1, n] + e[k-1, n+1]; e[, ] = 0; a[0] = 1; a[n_] := e[n, 0] + e[0, n]; Table[a[n], {n, 0, dim}] (* Jean-François Alcover, Jun 27 2013, translated and adapted from Maple *)
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == 0,
     Sum[b[u - j, o + j - 1, Mod[t + 1, 5]], {j, 1, u}],
     Sum[b[u + j - 1, o - j, Mod[t + 1, 5]], {j, 1, o}]]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Apr 21 2021, after Alois P. Heinz in A250283 *)
nmax = 30; CoefficientList[Series[1 + Sum[(x^(5 - k) * HypergeometricPFQ[{1}, {6/5 - k/5, 7/5 - k/5, 8/5 - k/5, 9/5 - k/5, 2 - k/5}, -x^5/3125])/(5 - k)!, {k, 0, 4}] / HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, -x^5/3125], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 21 2021 *)

@cached_function
def A(m,n):
    if n == 0: return 1
    s = -1 if m.divides(n) else 1
    t = [m*k for k in (0..(n-1)//m)]
    return s*add(binomial(n,k)*A(m,k) for k in t)
A181936 = lambda n: (-1)^int(is_odd(n//5))*A(5,n)
print([A181936(n) for n in (0..30)]) # Peter Luschny, Jan 24 2017

A249583 Number of permutations p of [n] such that p(i) > p(i+1) iff i == 2 (mod 3).

Original entry on oeis.org

1, 1, 1, 2, 5, 9, 40, 169, 477, 3194, 19241, 74601, 666160, 5216485, 25740261, 287316122, 2769073949, 16591655817, 222237912664, 2543467934449, 17929265150637, 280180369563194, 3712914075133121, 30098784753112329, 537546603651987424, 8094884285992309261

Offset: 0

Alois P. Heinz, Nov 01 2014

nonn

This is the (UDU)* version of 3-alternating permutations of [n], (U=Up, D=Down).

			a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 9: 13245, 14235, 15234, 23145, 24135, 25134, 34125, 35124, 45123.
a(6) = 40: 132465, 132564, 142365, 142563, 143562, 152364, 152463, 153462, 162354, 162453, 163452, 231465, 231564, 241365, 241563, 243561, 251364, 251463, 253461, 261354, 261453, 263451, 341265, 341562, 342561, 351264, 351462, 352461, 361254, 361452, 362451, 451263, 451362, 452361, 461253, 461352, 462351, 561243, 561342, 562341.
a(7) = 169: 1324657, 1324756, 1325647, ..., 6723514, 6724513, 6734512.

Alois P. Heinz, Table of n, a(n) for n = 0..500
J. M. Luck, On the frequencies of patterns of rises and falls, arXiv:1309.7764, 2013
Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page
R. P. Stanley, A survey of alternating permutations, arXiv:0912.4240, 2009

Cf. A178963 (i=0), A249402 (i=1).

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=2, add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..35);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == 2, Sum[b[u - j, o + j - 1, Mod[t+1, 3]], {j, 1, u}], Sum[b[u + j - 1, o - j, Mod[t+1, 3]], {j, 1, o}]]];
a[n_] := b[0, n, 0];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

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