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

Author

Peter Luschny, Dec 06 2023

Keywords

• nonn
• tabl

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

Cf. A094587, A000142 (T(n, 0)), A052849 (T(n, 1)), A000522 (T(n, n)), A007526 (T(n,n-1)), A038154 (T(n, n-2)), A355268 (T(n, n/2)), A367963(n) (T(2*n, n)/n!).

Cf. A001339 (row sums), A087208 (alternating row sums), A082030 (accumulated sums), A053482, A331689.

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)])

Formula

T(n, k) = A094587(n, k) * A000522(k).

T(n, k) = e * (n! / k!) * Gamma(k + 1, 1).

Sum_{k=0..n} T(n, k) * 2^(n - k) = A053482(n).

Sum_{k=0..n} T(n, k) * binomial(n, k) = A331689(n).

Recurrence: T(n, n) = T(n, n-1) + 1 starting with T(0, 0) = 1.

For k <> n: T(n, k) = n * T(n-1, k).