|
|
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
|
|
|
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
|
|
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)
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)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|