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A181937 André numbers. Square array A(n,k), n>=1, k>=0, read by antidiagonals, A(n,k) = n-alternating permutations of length k. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,10

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

Désiré André, Développement de sec x and tg x, C. R. Math. Acad. Sci. Paris 88 (1879), 965-979.

Désiré André, Mémoire sur les permutations alternées, J. Math. pur. appl., 7 (1881), 167-184.

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.

LINKS

Table of n, a(n) for n=1..76.

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

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 1 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

CROSSREFS

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

Sequence in context: A212382 A274835 A275069 * A233836 A214719 A259681

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 February 24 06:13 EST 2018. Contains 299597 sequences. (Running on oeis4.)