OFFSET
0,6
COMMENTS
For k >= 2 and p prime, T(p,k) == 0 (mod 4*p*(p-1)). - Mélika Tebni, Jan 20 2023
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
From Mélika Tebni, Jan 20 2023: (Start)
E.g.f. column k: (LambertW(-x) - log(1 + LambertW(-x)))^k / k!.
-Sum_{k=1..n/2} (-1)^k*T(n,k) = A071720(n+1), for n > 0.
-Sum_{k=1..n/2} (-1)^k*T(n,k) / (n-1) = A007830(n-2), for n > 1.
T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1, j-1)*A106828(j, k) for n > 0. (End)
EXAMPLE
Triangle T(n,k) begins:
1;
0;
0, 1;
0, 8;
0, 78, 3;
0, 944, 80;
0, 13800, 1810, 15;
0, 237432, 41664, 840;
0, 4708144, 1022252, 34300, 105;
0, 105822432, 27098784, 1286432, 10080;
0, 2660215680, 778128336, 47790540, 648900, 945;
...
MAPLE
c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
b(n-i)*binomial(n-1, i-1)*x*c(i), i=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);
MATHEMATICA
c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
b[n - i]*Binomial[n - 1, i - 1]*x*c[i], {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n/2}]][b[n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)
PROG
(PARI) \\ here AS1(n, k) gives associated Stirling numbers of 1st kind.
AS1(n, k)={(-1)^(n+k)*sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 1) )}
T(n, k) = {if(n==0, k==0, sum(j=k, n, n^(n-j)*binomial(n-1, j-1)*AS1(j, k)))} \\ Andrew Howroyd, Jan 20 2023
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Dec 31 2021
STATUS
approved