A347667 Triangle read by rows: T(n,k) = Sum_{j=0..k} binomial(n,j) * j! (0 <= k <= n).
1, 1, 2, 1, 3, 5, 1, 4, 10, 16, 1, 5, 17, 41, 65, 1, 6, 26, 86, 206, 326, 1, 7, 37, 157, 517, 1237, 1957, 1, 8, 50, 260, 1100, 3620, 8660, 13700, 1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601, 1, 10, 82, 586, 3610, 18730, 79210, 260650, 623530, 986410
Offset: 0
Examples
Triangle begins: 1; 1, 2; 1, 3, 5; 1, 4, 10, 16; 1, 5, 17, 41, 65; 1, 6, 26, 86, 206, 326; 1, 7, 37, 157, 517, 1237, 1957; 1, 8, 50, 260, 1100, 3620, 8660, 13700; ...
Programs
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Mathematica
T[n_, k_] := Sum[Binomial[n, j] j!, {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten