login
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)!).
4
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
0,5
FORMULA
T(n, k) = n! * [x^n] ((-LambertW(-x)/x - 1)^k / k!).
EXAMPLE
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;
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;
CROSSREFS
Cf. A124824 (row sums), A000272 (column 1), A045943 (subdiagonal).
Sequence in context: A228334 A114151 A243098 * A241981 A147723 A110518
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 28 2023
STATUS
approved