A181936 -id:A181936 - 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

A211212 4-alternating permutations of length 4n.

Original entry on oeis.org

1, 1, 69, 33661, 60376809, 288294050521, 3019098162602349, 60921822444067346581, 2159058013333667522020689, 125339574046311949415000577841, 11289082167259099068433198467575829, 1510335441937894173173702826484473600301

Offset: 0

Peter Luschny, Apr 04 2012

nonn

a(n) = A181985(4,n).

L. Carlitz, Permutations with prescribed pattern, Math. Nachr. 58 (1973), 31-53.
Peter Luschny, An old operation on sequences: the Seidel transform

Cf. A000364, A002115, A181985.

A211212 := proc(n) local E, dim, i, k; dim := 4*(n-1);
E := array(0..dim, 0..dim); E[0, 0] := 1;
for i from 1 to dim do
   if i mod 4 = 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, dim] end:
seq(A211212(i), i = 1..12);
A211212_list := proc(size) local E, S;
E := 2*exp(x*z)/(cosh(z)+cos(z));
S := z -> series(E, z, 4*(size+1));
seq((-1)^n*(4*n)!*subs(x=0, coeff(S(z), z, 4*n)), n=0..size-1) end:
A211212_list(12); # Peter Luschny, Jun 06 2016

A181985[n_, len_] := Module[{e, dim = n (len - 1)}, e[0, 0] = 1; For[i = 1, i <= dim, i++, If[Mod[i, n] == 0, e[i, 0] = 0; For[k = i - 1, k >= 0, k--, e[k, i - k] = e[k + 1, i - k - 1] + e[k, i - k - 1]], e[0, i] = 0; For[k = 1, k <= i, k++, e[k, i - k] = e[k - 1, i - k + 1] + e[k - 1, i - k]]]]; Table[e[0, n k], {k, 0, len - 1}]];
a[n_] := A181985[4, n + 1] // Last;
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Jun 29 2019 *)

# uses[A from A181936]
A211212 = lambda n: A(4,4*n)*(-1)^n
print([A211212(n) for n in (0..11)]) # Peter Luschny, Jan 24 2017

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(4*n,4*k) * a(n-k). - Ilya Gutkovskiy, Jan 27 2020
E.g.f.: 1/(cos(x/sqrt(2))*cosh(x/sqrt(2))) = 1 + 1*z^4/4! + 69*z^8/8! + 33661*z^12/12! + ... - Michael Wallner, Nov 17 2020
a(n) ~ 2^(10*n + 9/2) * n^(4*n + 1/2) / (cosh(Pi/2) * Pi^(4*n + 1/2) * exp(4*n)). - Vaclav Kotesovec, Nov 17 2020

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

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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A211212 4-alternating permutations of length 4n.

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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))).

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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)).

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))).