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A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

%I A367962 #25 Dec 07 2023 16:44:52
%S A367962 1,1,2,2,4,5,6,12,15,16,24,48,60,64,65,120,240,300,320,325,326,720,
%T A367962 1440,1800,1920,1950,1956,1957,5040,10080,12600,13440,13650,13692,
%U A367962 13699,13700,40320,80640,100800,107520,109200,109536,109592,109600,109601
%N A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).
%F A367962 T(n, k) = A094587(n, k) * A000522(k).
%F A367962 T(n, k) = e * (n! / k!) * Gamma(k + 1, 1).
%F A367962 Sum_{k=0..n} T(n, k) * 2^(n - k) = A053482(n).
%F A367962 Sum_{k=0..n} T(n, k) * binomial(n, k) = A331689(n).
%F A367962 Recurrence: T(n, n) = T(n, n-1) + 1 starting with T(0, 0) = 1.
%F A367962 For k <> n: T(n, k) = n * T(n-1, k).
%e A367962   [0]   1;
%e A367962   [1]   1,    2;
%e A367962   [2]   2,    4,    5;
%e A367962   [3]   6,   12,   15,   16;
%e A367962   [4]  24,   48,   60,   64,   65;
%e A367962   [5] 120,  240,  300,  320,  325,  326;
%e A367962   [6] 720, 1440, 1800, 1920, 1950, 1956, 1957;
%p A367962 T := (n, k) -> add(n!/j!, j = 0..k):
%p A367962 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A367962 Module[{n=1},NestList[Append[n#,1+Last[#]n++]&,{1},10]] (* or *)
%t A367962 Table[Sum[n!/j!,{j,0,k}],{n,0,10},{k,0,n}] (* _Paolo Xausa_, Dec 07 2023 *)
%o A367962 (SageMath)
%o A367962 def T(n, k): return sum(falling_factorial(n, n - j) for j in range(k + 1))
%o A367962 for n in range(9): print([T(n, k) for k in range(n + 1)])
%o A367962 (Python)
%o A367962 from functools import cache
%o A367962 @cache
%o A367962 def a_row(n: int) -> list[int]:
%o A367962     if n == 0: return [1]
%o A367962     row = a_row(n - 1) + [0]
%o A367962     for k in range(n): row[k] *= n
%o A367962     row[n] = row[n - 1] + 1
%o A367962     return row
%Y A367962 Cf. A094587, A000142 (T(n, 0)), A052849 (T(n, 1)), A000522 (T(n, n)), A007526 (T(n,n-1)), A038154 (T(n, n-2)), A355268 (T(n, n/2)), A367963(n) (T(2*n, n)/n!).
%Y A367962 Cf. A001339 (row sums), A087208 (alternating row sums), A082030 (accumulated sums), A053482, A331689.
%K A367962 nonn,tabl
%O A367962 0,3
%A A367962 _Peter Luschny_, Dec 06 2023