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

%I A371686 #11 Apr 07 2024 03:50:37
%S A371686 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,
%T A371686 975,1956,1957,1,14,105,560,2275,6846,13699,13700,1,16,140,896,4550,
%U A371686 18256,54796,109600,109601,1,18,180,1344,8190,41076,164388,493200,986409,986410
%N A371686 Triangle read by rows: T(n, k) = e * binomial(n, k) * Gamma(k + 1, 1).
%F A371686 T(n, k) = (n! / (n - k)!)*(Sum_{j = 0..k} (1 / j!)). - _Detlef Meya_, Apr 06 2024
%e A371686 Triangle starts:
%e A371686   [0] 1;
%e A371686   [1] 1,  2;
%e A371686   [2] 1,  4,   5;
%e A371686   [3] 1,  6,  15,  16;
%e A371686   [4] 1,  8,  30,  64,   65;
%e A371686   [5] 1, 10,  50, 160,  325,  326;
%e A371686   [6] 1, 12,  75, 320,  975, 1956,  1957;
%e A371686   [7] 1, 14, 105, 560, 2275, 6846, 13699, 13700;
%p A371686 T := (n, k) -> binomial(n, k)*GAMMA(k + 1, 1)*exp(1):
%p A371686 seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9);
%t A371686 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 *)
%Y A371686 Cf. A000522 (main diagonal), A007526 (subdiagonal), A010842 (row sums), A000142 and A133942 (alternating row sums), A367963 (central terms).
%K A371686 nonn,tabl
%O A371686 0,3
%A A371686 _Peter Luschny_, Apr 03 2024