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A366159 Triangle read by rows: T(n, k) = Sum_{i=0..k-2} (-1)^(i+2) * (k-i-1)^n * binomial(k,i).

%I A366159 #16 Oct 02 2023 13:44:55
%S A366159 1,1,5,1,13,23,1,29,121,119,1,61,479,1081,719,1,125,1681,6719,10081,
%T A366159 5039,1,253,5543,35281,90719,100801,40319,1,509,17641,168839,665281,
%U A366159 1239839,1088641,362879,1,1021,54959,763561,4339439,12096001,17539199,12700801,3628799
%N A366159 Triangle read by rows: T(n, k) = Sum_{i=0..k-2} (-1)^(i+2) * (k-i-1)^n * binomial(k,i).
%H A366159 Michael De Vlieger, <a href="/A366159/b366159.txt">Table of n, a(n) for n = 2..11176</a> (rows 2..150, flattened)
%H A366159 Dmitry N. Kozlov, <a href="https://arxiv.org/abs/2309.17142">Stirling complexes</a>, arXiv:2309.17142 [math.CO], 2023.
%e A366159 Triangle begins:
%e A366159   1;
%e A366159   1,    5;
%e A366159   1,   13,    23;
%e A366159   1,   29,   121,    119;
%e A366159   1,   61,   479,   1081,     719;
%e A366159   1,  125,  1681,   6719,   10081,     5039;
%e A366159   1,  253,  5543,  35281,   90719,   100801,    40319;
%e A366159   1,  509, 17641, 168839,  665281,  1239839,  1088641,   362879;
%e A366159   1, 1021, 54959, 763561, 4339439, 12096001, 17539199, 12700801, 3628799;
%e A366159   ...
%t A366159 Table[Sum[(-1)^(i + 2)*(k - i - 1)^n*Binomial[k, i], {i, 0, k - 2} ], {n, 2, 10}, {k, 2, n}] // Flatten (* _Michael De Vlieger_, Oct 02 2023 *)
%o A366159 (PARI) T(n, k) = sum(i=0, k-2, (-1)^(i+2) * (k-i-1)^n * binomial(k,i));
%o A366159 tabl(nn) = for (n=2, nn, for (k=2, n, print1(T(n,k), ", ")));
%Y A366159 Cf. A000012 (col 2), A036563 (col 3), A033312 (right border).
%Y A366159 Cf. A105060.
%K A366159 nonn,tabl
%O A366159 2,3
%A A366159 _Michel Marcus_, Oct 02 2023