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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. 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

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

REFERENCES

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

LINKS

Alois P. Heinz, Antidiagonals k = 0..140, flattened

Désiré André, Développement de séc x and 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]

EXAMPLE

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]

MAPLE

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;

MATHEMATICA

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 *)

PROG

(Sage)

@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

(Julia) # 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

CROSSREFS

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

Sequence in context: A212382 A274835 A275069 * A233836 A214719 A327858

Adjacent sequences:  A181934 A181935 A181936 * A181938 A181939 A181940

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Apr 03 2012

STATUS

approved

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Last modified October 15 12:28 EDT 2019. Contains 328026 sequences. (Running on oeis4.)