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A181985 Generalized Euler numbers. Square array A(n,k), n >= 1, k >= 0, read by antidiagonals. A(n,k) = n-alternating permutations of length n*k.

%I A181985 #35 Mar 04 2020 16:49:19
%S A181985 1,1,1,1,1,1,1,1,5,1,1,1,19,61,1,1,1,69,1513,1385,1,1,1,251,33661,
%T A181985 315523,50521,1,1,1,923,750751,60376809,136085041,2702765,1,1,1,3431,
%U A181985 17116009,11593285251,288294050521,105261234643,199360981,1
%N A181985 Generalized Euler numbers. Square array A(n,k), n >= 1, k >= 0, read by antidiagonals. A(n,k) = n-alternating permutations of length n*k.
%C A181985 For an integer n > 0, a permutation s = s_1...s_k is an n-alternating permutation if it has the property that s_i < s_{i+1} if and only if n divides i.
%C A181985 The classical Euler numbers count 2-alternating permutations of length 2n.
%C A181985 Ludwig Seidel gave in 1877 an efficient algorithm to compute the coefficients of sec which carries immediately over to the computation of the generalized Euler numbers (see the Maple script).
%H A181985 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%H A181985 Ludwig Seidel, <a href="https://babel.hathitrust.org/cgi/pt?id=hvd.32044092897461&amp;view=1up&amp;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 A181985 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 A181985 n\k [0][1]  [2]       [3]            [4]                 [5]
%e A181985 [1]  1, 1,   1,        1,             1,                   1
%e A181985 [2]  1, 1,   5,       61,          1385,               50521  [A000364]
%e A181985 [3]  1, 1,  19,     1513,        315523,           136085041  [A002115]
%e A181985 [4]  1, 1,  69,    33661,      60376809,        288294050521  [A211212]
%e A181985 [5]  1, 1, 251,   750751,   11593285251,     613498040952501
%e A181985 [6]  1, 1, 923, 17116009, 2301250545971, 1364944703949044401
%e A181985        [A030662][A211213]   [A181991]
%e A181985 The (n,n)-diagonal is A181992.
%p A181985 A181985_list := proc(n, len) local E, dim, i, k;
%p A181985 dim := n*(len-1); E := array(0..dim, 0..dim); E[0, 0] := 1;
%p A181985 for i from 1 to dim do
%p A181985    if i mod n = 0 then E[i, 0] := 0 ;
%p A181985       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 A181985    else E[0, i] := 0;
%p A181985       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 A181985    fi od;
%p A181985 seq(E[0, n*k], k=0..len-1) end:
%p A181985 for n from 1 to 6 do print(A181985_list(n, 6)) od;
%t A181985 nmax = 9; 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}]]; t = Table[A181985[n, nmax], {n, 1, nmax}]; a[n_, k_] := t[[n, k+1]]; Table[a[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jun 27 2013, translated and adapted from Maple *)
%o A181985 (Sage)
%o A181985 def A181985(m, n):
%o A181985     shapes = ([x*m for x in p] for p in Partitions(n))
%o A181985     return (-1)^n*sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
%o A181985 for m in (1..6): print([A181985(m, n) for n in (0..7)]) # _Peter Luschny_, Aug 10 2015
%Y A181985 Cf. A181937, A000364, A002115, A030662, A211212, A211213, A181991, A181992.
%K A181985 nonn,tabl
%O A181985 1,9
%A A181985 _Peter Luschny_, Apr 04 2012