A371686 Triangle read by rows: T(n, k) = e * binomial(n, k) * Gamma(k + 1, 1).

1, 1, 2, 1, 4, 5, 1, 6, 15, 16, 1, 8, 30, 64, 65, 1, 10, 50, 160, 325, 326, 1, 12, 75, 320, 975, 1956, 1957, 1, 14, 105, 560, 2275, 6846, 13699, 13700, 1, 16, 140, 896, 4550, 18256, 54796, 109600, 109601, 1, 18, 180, 1344, 8190, 41076, 164388, 493200, 986409, 986410

Offset

0,3

Links

Table of n, a(n) for n=0..54.

Formula

T(n, k) = (n! / (n - k)!)*(Sum_{j = 0..k} (1 / j!)). - Detlef Meya, Apr 06 2024

Example

Triangle starts:
  [0] 1;
  [1] 1,  2;
  [2] 1,  4,   5;
  [3] 1,  6,  15,  16;
  [4] 1,  8,  30,  64,   65;
  [5] 1, 10,  50, 160,  325,  326;
  [6] 1, 12,  75, 320,  975, 1956,  1957;
  [7] 1, 14, 105, 560, 2275, 6846, 13699, 13700;

Maple

T := (n, k) -> binomial(n, k)*GAMMA(k + 1, 1)*exp(1):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9);

Mathematica

T[n_, k_]:=(n!/(n-k)!)*Sum[1/j!, {j, 0, k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, Apr 06 2024 *)

Crossrefs

Cf. A000522 (main diagonal), A007526 (subdiagonal), A010842 (row sums), A000142 and A133942 (alternating row sums), A367963 (central terms).

Sequence in context: A124959 A081281 A108198 * A321000 A121289 A211561

Adjacent sequences: A371683 A371684 A371685 * A371687 A371688 A371689

Keyword

nonn,tabl

Author

Peter Luschny, Apr 03 2024

Status

approved