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A347667 Triangle read by rows: T(n,k) = Sum_{j=0..k} binomial(n,j) * j! (0 <= k <= n).

%I A347667 #5 Sep 10 2021 19:22:57
%S A347667 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,
%T A347667 1237,1957,1,8,50,260,1100,3620,8660,13700,1,9,65,401,2081,8801,28961,
%U A347667 69281,109601,1,10,82,586,3610,18730,79210,260650,623530,986410
%N A347667 Triangle read by rows: T(n,k) = Sum_{j=0..k} binomial(n,j) * j! (0 <= k <= n).
%e A347667 Triangle begins:
%e A347667   1;
%e A347667   1, 2;
%e A347667   1, 3,  5;
%e A347667   1, 4, 10,  16;
%e A347667   1, 5, 17,  41,   65;
%e A347667   1, 6, 26,  86,  206,  326;
%e A347667   1, 7, 37, 157,  517, 1237, 1957;
%e A347667   1, 8, 50, 260, 1100, 3620, 8660, 13700;
%e A347667   ...
%t A347667 T[n_, k_] := Sum[Binomial[n, j] j!, {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%Y A347667 T(n,n) = A000522, T(2*n,n) = A066211.
%Y A347667 Cf. A008279, A008949, A111063, A121662, A285268.
%K A347667 nonn,tabl
%O A347667 0,3
%A A347667 _Ilya Gutkovskiy_, Sep 10 2021