A360177 Triangle read by rows. T(n, k) = 1 if n = k, otherwise T(n, k) = Sum_{j=0..k-1} (-1)^(j - k - 1) * (n + j + 1)^(n-1) / (j! * (k - 1 - j)!).

1, 0, 1, 0, 3, 1, 0, 16, 9, 1, 0, 125, 91, 18, 1, 0, 1296, 1105, 295, 30, 1, 0, 16807, 15961, 5160, 725, 45, 1, 0, 262144, 269297, 99631, 17290, 1505, 63, 1, 0, 4782969, 5217031, 2135070, 431221, 46970, 2786, 84, 1, 0, 100000000, 114358881, 50631967, 11477046, 1471701, 110250, 4746, 108, 1

Offset: 0

Author

Peter Luschny, Jan 28 2023

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,       1;
[2] 0,       3,       1;
[3] 0,      16,       9,       1;
[4] 0,     125,      91,      18,      1;
[5] 0,    1296,    1105,     295,     30,     1;
[6] 0,   16807,   15961,    5160,    725,    45,     1;
[7] 0,  262144,  269297,   99631,  17290,  1505,    63,  1;
[8] 0, 4782969, 5217031, 2135070, 431221, 46970,  2786, 84, 1;
		

Crossrefs

Cf. A124824 (row sums), A000272 (column 1), A045943 (subdiagonal).

Programs

Maple

A360177 := (n, k) -> if n = k then 1 else
add((-1)^(u-k-1)*(n+u+1)^(n-1)/(u!*(k-1-u)!), u = 0.. k-1) fi:
for n from 0 to 8 do seq(A360177(n, k), k = 0..n) od;
# Alternative:
egf := k -> (-LambertW(-x)/x - 1)^k / k!:
ser := k -> series(egf(k), x, 22): T := (n, k) -> n!*coeff(ser(k), x, n):
for n from 0 to 8 do print(seq(T(n, k), k = 0..n)) od;

Formula

T(n, k) = n! * [x^n] ((-LambertW(-x)/x - 1)^k / k!).