A178964 -id:A178964 - 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

A178963 E.g.f.: (3+2sqrt(3)exp(x/2)sin(sqrt(3)x/2))/(exp(-x)+2exp(x/2)cos(sqrt(3)*x/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 19, 99, 477, 1513, 11259, 74601, 315523, 3052323, 25740261, 136085041, 1620265923, 16591655817, 105261234643, 1488257158851, 17929265150637, 132705221399353, 2172534146099019, 30098784753112329, 254604707462013571, 4736552519729393091, 74180579084559895221, 705927677520644167681, 14708695606607601165843, 256937013876000351610089, 2716778010767155313771539

Offset: 0

N. J. A. Sloane, Dec 31 2010

nonn

According to Mendes and Remmel, p. 56, this is the e.g.f. for 3-alternating permutations.

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 preprint arXiv:1309.7764 [cond-mat.stat-mech], 2013-2014.
Peter Luschny, An old operation on sequences: the Seidel transform.
Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page.
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), A181936 (m=5).
Cf. A181937, A002115.
Cf. A249402, A249583 (alternative definitions of 3-alternating permutations).

A178963_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 3 = 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:
A178963_list(30);  # Peter Luschny, Apr 02 2012
# Alternatively, using a bivariate exponential generating function:
A178963 := proc(n) local g, p, q;
g := (x,z) -> 3*exp(x*z)/(exp(z)+2*exp(-z/2)*cos(z*sqrt(3)/2));
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)^floor(n/3)*p(n,q(n,3)) end:
seq(A178963(i),i=0..30); # Peter Luschny, Jun 06 2012
# third Maple program:
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(n, 0, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 29 2014

max = 30; f[x_] := (E^x*(2*Sqrt[3]*E^(x/2)*Sin[(Sqrt[3]*x)/2] + 3))/(2*E^((3*x)/2)*Cos[(Sqrt[3]*x)/2] + 1); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! // Simplify (* Jean-François Alcover, Sep 16 2013 *)

# uses[A from A181936]
A178963 = lambda n: (-1)^int(is_odd(n//3))*A(3,n)
print([A178963(n) for n in (0..30)]) # Peter Luschny, Jan 24 2017

a(3*n) = A002115(n). - Peter Luschny, Aug 02 2012

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

A281587 Triangle read by rows, e.g.f. exp(xz)(2*(exp(z)+1)/(cosh(z)+cos(z))-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, -1, 4, 6, 4, 1, -4, -5, 10, 10, 5, 1, -14, -24, -15, 20, 15, 6, 1, -34, -98, -84, -35, 35, 21, 7, 1, 69, -272, -392, -224, -70, 56, 28, 8, 1, 496, 621, -1224, -1176, -504, -126, 84, 36, 9, 1, 2896, 4960, 3105, -4080, -2940, -1008, -210, 120, 45, 10, 1

Offset: 0

Peter Luschny, Feb 01 2017

			Triangle starts:
[   1]
[   1,   1]
[   1,   2,   1]
[   1,   3,   3,   1]
[  -1,   4,   6,   4,  1]
[  -4,  -5,  10,  10,  5,  1]
[ -14, -24, -15,  20, 15,  6, 1]
[ -34, -98, -84, -35, 35, 21, 7, 1]

Cf. A130595 (m=1), A109449 (m=2), A215064 (m=3), A281587 (m=4).

Cf. A178964 (col. 0), A281588 (col. 1).

A281587_row := proc(n) exp(x*z)*(2*(exp(z)+1)/(cosh(z)+cos(z))-1);
n!*coeff(series(%,z,n+1),z,n); PolynomialTools:-CoefficientList(%,x) end:
for n from 0 to 12 do A281587_row(n) od;

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.

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A178963 E.g.f.: (3+2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2))/(exp(-x)+2*exp(x/2)*cos(sqrt(3)*x/2)).

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A181936 Number of 5-alternating permutations.

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A281587 Triangle read by rows, e.g.f. exp(x*z)*(2*(exp(z)+1)/(cosh(z)+cos(z))-1).

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Maple

A178963 E.g.f.: (3+2sqrt(3)exp(x/2)sin(sqrt(3)x/2))/(exp(-x)+2exp(x/2)cos(sqrt(3)*x/2)).

A281587 Triangle read by rows, e.g.f. exp(xz)(2*(exp(z)+1)/(cosh(z)+cos(z))-1).