OFFSET
0,3
COMMENTS
T(n, k) = [x^k] series[ H(n - k, x) ]
Here H(n,x) = E(n,x)*(1+x)/(1-x)^(n+1) where E(n,x) are the Eulerian polynomials, E(0,x) = 1 and E(n,x) = sum_{k=0^{n-1}} W_{n,k} x^k for n > 0. W_{n,k} as in DLMF Table 26.14.1.
LINKS
EXAMPLE
1
1, 2
1, 3, 2
1, 5, 5, 2
1, 9, 13, 7, 2
1, 17, 35, 25, 9, 2
1, 33, 97, 91, 41, 11, 2
MAPLE
MATHEMATICA
e[0, _] = 1; e[n_, x_] := e[n, x] = x(1-x) D[e[n-1, x], x] + e[n-1, x] (1 + (n-1)x);
h[n_, x_] := e[n, x] (1+x)/(1-x)^(n+1);
T[n_, k_] := SeriesCoefficient[h[n-k, x], {x, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-François Alcover, Jun 17 2019, from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Aug 02 2010
STATUS
approved