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%I #58 Nov 22 2023 12:08:48
%S 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,
%T 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,
%U 1,1,20,69,1513,50521,1,1,1,1,1,1,1,6,55,496,11259,353792
%N André numbers. Square array A(n,k), n>=2, k>=0, read by antidiagonals upwards, A(n,k) = n-alternating permutations of length k.
%C 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.
%D Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page.
%H Alois P. Heinz, <a href="/A181937/b181937.txt">Antidiagonals k = 0..140, flattened</a>
%H Désiré André, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k30457/f961.image">Développement de séc x et de tang x</a>, C. R. Math. Acad. Sci. Paris 88 (1879), 965-967.
%H Désiré André, <a href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1881_3_7_A10_0.pdf">Sur les permutations alternées</a>, J. Math. pur. appl., 7 (1881), 167-184.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%H Ludwig Seidel, <a href="https://babel.hathitrust.org/cgi/pt?id=hvd.32044092897461&view=1up&seq=176">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the <a href="https://www.hathitrust.org/accessibility">HATHI TRUST Digital Library</a>]
%H Ludwig Seidel, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1877_0157-0187.pdf">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through <a href="https://de.wikipedia.org/wiki/ZOBODAT">ZOBODAT</a>]
%e n\k [0][1][2][3][4] [5] [6] [7] [8] [9] [10] [11]
%e [1] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 [A000012]
%e [2] 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792 [A000111]
%e [3] 1, 1, 1, 1, 3, 9, 19, 99, 477, 1513, 11259, 74601 [A178963]
%e [4] 1, 1, 1, 1, 1, 4, 14, 34, 69, 496, 2896, 11056 [A178964]
%e [5] 1, 1, 1, 1, 1, 1, 5, 20, 55, 125, 251, 2300 [A181936]
%e [6] 1, 1, 1, 1, 1, 1, 1, 6, 27, 83, 209, 461 [A250283]
%p A181937_list := proc(n, len) local E,dim,i,k; # Seidel's boustrophedon transform
%p dim := len-1; E := array(0..dim, 0..dim); E[0,0] := 1;
%p for i from 1 to dim do
%p if i mod n = 0 then E[i,0] := 0 ;
%p 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;
%p else E[0,i] := 0;
%p 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;
%p fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..dim)] end:
%p for n from 2 to 6 do print(A181937_list(n,12)) od;
%t 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 *)
%t 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 *)
%o (Sage)
%o @cached_function
%o def A(m, n):
%o if n == 0: return 1
%o s = -1 if m.divides(n) else 1
%o t = [m*k for k in (0..(n-1)//m)]
%o return s*add(binomial(n, k)*A(m, k) for k in t)
%o A181937_row = lambda m, n: (-1)^int(is_odd(n//m))*A(m, n)
%o for n in (1..6): print([A181937_row(n, k) for k in (0..20)]) # _Peter Luschny_, Feb 06 2017
%o (Julia) # Signed version.
%o using Memoize
%o @memoize function André(m, n)
%o n ≤ 0 && return 1
%o r = range(0, stop=n-1, step=m)
%o S = sum(binomial(n, k) * André(m, k) for k in r)
%o n % m == 0 ? -S : S
%o end
%o for m in 1:8 println([André(m, n) for n in 0:11]) end # _Peter Luschny_, Feb 09 2019
%Y Cf. A000111, A178963, A178964, A181936, A250283, A250284, A250285, A250286, A250287.
%K nonn,tabl
%O 0,10
%A _Peter Luschny_, Apr 03 2012
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