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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 May 26 15:27 EDT 2020. Contains 334626 sequences. (Running on oeis4.)