A361042
Triangle read by rows: T(n, k) = Sum_{j=0..n} j! * binomial(n - j, n - k).
Original entry on oeis.org
1, 1, 2, 1, 3, 4, 1, 4, 7, 10, 1, 5, 11, 17, 34, 1, 6, 16, 28, 51, 154, 1, 7, 22, 44, 79, 205, 874, 1, 8, 29, 66, 123, 284, 1079, 5914, 1, 9, 37, 95, 189, 407, 1363, 6993, 46234, 1, 10, 46, 132, 284, 596, 1770, 8356, 53227, 409114, 1, 11, 56, 178, 416, 880, 2366, 10126, 61583, 462341, 4037914
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 2;
[2] 1, 3, 4;
[3] 1, 4, 7, 10;
[4] 1, 5, 11, 17, 34;
[5] 1, 6, 16, 28, 51, 154;
[6] 1, 7, 22, 44, 79, 205, 874;
[7] 1, 8, 29, 66, 123, 284, 1079, 5914;
[8] 1, 9, 37, 95, 189, 407, 1363, 6993, 46234;
[9] 1, 10, 46, 132, 284, 596, 1770, 8356, 53227, 409114.
A371686
Triangle read by rows: T(n, k) = e * binomial(n, k) * Gamma(k + 1, 1).
Original entry on oeis.org
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
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;
-
T := (n, k) -> binomial(n, k)*GAMMA(k + 1, 1)*exp(1):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9);
-
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 *)