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A355266 Triangle read by rows, T(n, k) = (-1)^(n-k)*Bell(k)*Stirling1(n+1, k+1), for 0 <= k <= n.

%I A355266 #10 Jul 07 2022 13:21:20
%S A355266 1,1,1,2,3,2,6,11,12,5,24,50,70,50,15,120,274,450,425,225,52,720,1764,
%T A355266 3248,3675,2625,1092,203,5040,13068,26264,33845,29400,16744,5684,877,
%U A355266 40320,109584,236248,336420,336735,235872,110838,31572,4140
%N A355266 Triangle read by rows, T(n, k) = (-1)^(n-k)*Bell(k)*Stirling1(n+1, k+1), for 0 <= k <= n.
%F A355266 T(n, k) = A000110(k) * A130534(n, k).
%F A355266 Sum_{k=0..n} T(n, k) = n!*Laguerre(n, -1) = A002720(n).
%F A355266 Sum_{k=0..n} (-1)^k*T(n, k) = !n = n!*A053557(n)/A053556(n) = A000166(n).
%e A355266 Triangle T(n, k) begins:
%e A355266 [0]    1;
%e A355266 [1]    1,      1;
%e A355266 [2]    2,      3,     2;
%e A355266 [3]    6,     11,    12,      5;
%e A355266 [4]   24,     50,    70,     50,    15;
%e A355266 [5]  120,    274,   450,    425,   225,     52;
%e A355266 [6]  720,   1764,  3248,   3675,  2625,   1092,  203;
%e A355266 [7] 5040,  13068, 26264,  33845, 29400,  16744, 5684, 877;
%p A355266 T := (n, k) -> (-1)^(n-k)*combinat:-bell(k)*Stirling1(n+1, k+1):
%p A355266 seq(seq(T(n, k), k = 0..n), n = 0..8);
%o A355266 (Python)
%o A355266 from functools import cache
%o A355266 @cache
%o A355266 def b(n: int, k=0):
%o A355266     return int(n < 1) or k * b(n - 1, k) + b(n - 1, k + 1)
%o A355266 @cache
%o A355266 def s(n: int) -> list[int]:
%o A355266     if n == 0: return [1]
%o A355266     row = [0] + s(n - 1)
%o A355266     for k in range(1, n): row[k] = row[k] + (n - 1) * row[k + 1]
%o A355266     return row
%o A355266 def A355266_row(n):
%o A355266     return [s * b(k - 1) for k, s in enumerate(s(n + 1))][1:]
%o A355266 for n in range(9): print(A355266_row(n))
%Y A355266 Cf. A002720 (row sums), A000166 (alternating row sums), A000110 (main diagonal), A000142 (column 0).
%Y A355266 Cf. A105479, A000254, A130534, A053557, A053556.
%K A355266 nonn,tabl
%O A355266 0,4
%A A355266 _Peter Luschny_ and _Mélika Tebni_, Jul 05 2022