login
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

%I #35 Mar 04 2020 16:49:19

%S 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 315523,50521,1,1,1,923,750751,60376809,136085041,2702765,1,1,1,3431,

%U 17116009,11593285251,288294050521,105261234643,199360981,1

%N 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 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 The classical Euler numbers count 2-alternating permutations of length 2n.

%C 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 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&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 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]

%e [1] 1, 1, 1, 1, 1, 1

%e [2] 1, 1, 5, 61, 1385, 50521 [A000364]

%e [3] 1, 1, 19, 1513, 315523, 136085041 [A002115]

%e [4] 1, 1, 69, 33661, 60376809, 288294050521 [A211212]

%e [5] 1, 1, 251, 750751, 11593285251, 613498040952501

%e [6] 1, 1, 923, 17116009, 2301250545971, 1364944703949044401

%e [A030662][A211213] [A181991]

%e The (n,n)-diagonal is A181992.

%p A181985_list := proc(n, len) local E, dim, i, k;

%p dim := n*(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;

%p seq(E[0, n*k], k=0..len-1) end:

%p for n from 1 to 6 do print(A181985_list(n, 6)) od;

%t 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 (Sage)

%o def A181985(m, n):

%o shapes = ([x*m for x in p] for p in Partitions(n))

%o return (-1)^n*sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)

%o for m in (1..6): print([A181985(m, n) for n in (0..7)]) # _Peter Luschny_, Aug 10 2015

%Y Cf. A181937, A000364, A002115, A030662, A211212, A211213, A181991, A181992.

%K nonn,tabl

%O 1,9

%A _Peter Luschny_, Apr 04 2012