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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. 8
 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, 315523, 50521, 1, 1, 1, 923, 750751, 60376809, 136085041, 2702765, 1, 1, 1, 3431, 17116009, 11593285251, 288294050521, 105261234643, 199360981, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS 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. The classical Euler numbers count 2-alternating permutations of length 2n. 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). LINKS 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] [1]  1, 1,   1,        1,             1,                   1 [2]  1, 1,   5,       61,          1385,               50521  [A000364] [3]  1, 1,  19,     1513,        315523,           136085041  [A002115] [4]  1, 1,  69,    33661,      60376809,        288294050521  [A211212] [5]  1, 1, 251,   750751,   11593285251,     613498040952501 [6]  1, 1, 923, 17116009, 2301250545971, 1364944703949044401        [A030662][A211213]   [A181991] The (n,n)-diagonal is A181992. MAPLE A181985_list := proc(n, len) local E, dim, i, k; dim := n*(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; seq(E[0, n*k], k=0..len-1) end: for n from 1 to 6 do print(A181985_list(n, 6)) od; MATHEMATICA 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 *) PROG (Sage) def A181985(m, n):     shapes = ([x*m for x in p] for p in Partitions(n))     return (-1)^n*sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes) for m in (1..6): print([A181985(m, n) for n in (0..7)]) # Peter Luschny, Aug 10 2015 CROSSREFS Cf. A181937, A000364, A002115, A030662, A211212, A211213, A181991, A181992. Sequence in context: A285486 A230368 A256690 * A304320 A130511 A320410 Adjacent sequences:  A181982 A181983 A181984 * A181986 A181987 A181988 KEYWORD nonn,tabl AUTHOR Peter Luschny, Apr 04 2012 STATUS approved

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Last modified January 22 10:23 EST 2021. Contains 340362 sequences. (Running on oeis4.)