A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).
1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601
Offset: 0
Examples
[0] 1; [1] 1, 2; [2] 2, 4, 5; [3] 6, 12, 15, 16; [4] 24, 48, 60, 64, 65; [5] 120, 240, 300, 320, 325, 326; [6] 720, 1440, 1800, 1920, 1950, 1956, 1957;
Crossrefs
Programs
-
Maple
T := (n, k) -> add(n!/j!, j = 0..k): seq(seq(T(n, k), k = 0..n), n = 0..9);
-
Mathematica
Module[{n=1},NestList[Append[n#,1+Last[#]n++]&,{1},10]] (* or *) Table[Sum[n!/j!,{j,0,k}],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 07 2023 *) -
Python
from functools import cache @cache def a_row(n: int) -> list[int]: if n == 0: return [1] row = a_row(n - 1) + [0] for k in range(n): row[k] *= n row[n] = row[n - 1] + 1 return row -
SageMath
def T(n, k): return sum(falling_factorial(n, n - j) for j in range(k + 1)) for n in range(9): print([T(n, k) for k in range(n + 1)])