OFFSET
0,6
COMMENTS
The OEIS values correct the values b(n,k) in the Laradji-Umar Table 2.1 in column k=2. Note that the row sums (meaning: sums up to the diagonal of the triangle) in Table 2.1 in the article are also incorrect.
There were typos in the column (k=2) of the original article. The entry 94 should be 166 and the entry 784 should be 856, which have been corrected. Unlike most triangles the off-diagonal terms are not 0 because T(n, n)= T(n, n+k) for all nonnegative k which is obvious from the definition.
LINKS
A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.
FORMULA
T(n,k) = T(n-1,k) + 3(n-1)T(n-2,k) + ... +(k+1)(n-1)(n-2)...(n-k+1)T(n-k,k) if k<=n.
T(n,k) = T(n,n) if k>n, not part of the triangle.
T(n,0) = T(n,1) = 1.
T(n,n) = A144085(n). (Diagonal)
G.f.: exp(x+(3x^2)/2+ ... +((k+1)x^k)/k).
EXAMPLE
T(3,2) = 10 because there are 10 subpermutations on {1,2,3}, each of whose orbit is of size at most 2, and without fixed points, namely: Empty map, (1,2) --> (2,1), (1,3) --> (3,1) (2,3) --> (3,2), 1-->2, 1-->3, 2-->1, 2-->3, 3-->1, 3-->2.
Triangle starts:
1;
1, 1;
1, 1, 4;
1, 1, 10, 18;
1, 1, 46, 78, 108;
1, 1, 166, 486, 636, 780;
...
MAPLE
A261762 := proc(n, k)
if k = 0 then
1;
else
if k < 1 then
g := 1;
elif k < 2 then
g := exp(x) ;
else
g := exp(x+add((j+1)*x^j/j, j=2..k)) ;
fi;
coeftayl(g, x=0, n) *n! ;
end if;
end proc:
seq(seq( A261762(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Nov 04 2015
MATHEMATICA
T[n_, k_] := SeriesCoefficient[ Exp[ x + Sum[ (j+1)*x^j/j, {j, 2, k}]], {x, 0, n}] * n!; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 13 2017 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Samira Stitou, Sep 21 2015
EXTENSIONS
More terms from Alois P. Heinz, Oct 07 2015
STATUS
approved