OFFSET
0,3
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
From Mélika Tebni, Mar 23 2023: (Start)
E.g.f. of column k: (W(-x)-log(1 + W(-x)))^k / (exp(W(-x))*k!), W(x) the Lambert W-function.
T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1,j-1)*A136394(j,k), for n > 0.
T(n,k) = Sum_{j=k..n} (n-j+1)^(n-j-1)*binomial(n,j)*A350452(j,k).
Sum_{k=0..n/2} (k+1)*T(n,k) = A190314(n), for n > 0.
Sum_{k=0..n/2} 2^k*T(n,k) = A217701(n). (End)
EXAMPLE
Triangle T(n,k) begins:
1;
1;
3, 1;
16, 11;
125, 128, 3;
1296, 1734, 95;
16807, 27409, 2425, 15;
262144, 499400, 61054, 945;
4782969, 10346328, 1605534, 42280, 105;
100000000, 240722160, 44981292, 1706012, 11025;
2357947691, 6222652233, 1351343346, 67291910, 763875, 945;
...
MAPLE
c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
t:= proc(n) option remember; n^(n-1) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
b(n-i)*binomial(n-1, i-1)*(c(i)*x+t(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);
# second Maple program:
egf := k-> (LambertW(-x)-log(1+LambertW(-x)))^k/(exp(LambertW(-x))*k!):
A350446 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350446(n, k), k=0..n/2)), n=0..10); # Mélika Tebni, Mar 23 2023
MATHEMATICA
c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
t[n_] := t[n] = n^(n - 1);
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
b[n - i]*Binomial[n - 1, i - 1]*(c[i]*x + t[i]), {i, 1, n}]]];
T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Dec 31 2021
STATUS
approved