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A350262 Triangle read by rows. T(n, k) = B(n, n - k + 1) where B(n, k) = k^n * BellPolynomial(n, -1/k) for k > 0, if k = 0 then B(n, k) = k^n.

%I A350262 #8 Dec 30 2021 07:23:14
%S A350262 1,-1,-1,-2,-1,0,-5,-1,1,1,21,25,19,9,1,1103,674,343,128,23,-2,29835,
%T A350262 15211,6551,2133,379,-25,-9,739751,331827,123821,33479,3603,-1549,
%U A350262 -583,-9,16084810,5987745,1619108,120865,-174114,-112975,-32600,-3087,50
%N A350262 Triangle read by rows. T(n, k) = B(n, n - k + 1) where B(n, k) = k^n * BellPolynomial(n, -1/k) for k > 0, if k = 0 then B(n, k) = k^n.
%e A350262 [0]        1
%e A350262 [1]       -1,      -1
%e A350262 [2]       -2,      -1,       0
%e A350262 [3]       -5,      -1,       1,      1
%e A350262 [4]       21,      25,      19,      9,       1
%e A350262 [5]     1103,     674,     343,    128,      23,      -2
%e A350262 [6]    29835,   15211,    6551,   2133,     379,     -25,     -9
%e A350262 [7]   739751,  331827,  123821,  33479,    3603,   -1549,   -583,    -9
%e A350262 [8] 16084810, 5987745, 1619108, 120865, -174114, -112975, -32600, -3087, 50
%p A350262 B := (n, k) -> ifelse(k = 0, k^n, k^n * BellB(n, -1/k)):
%p A350262 A350262 := (n, k) -> B(n, n - k + 1):
%p A350262 seq(seq(A350262(n, k), k = 0..n), n = 0..8);
%t A350262 B[n_, k_] := If[k == 0, k^n, k^n BellB[n, -1/k]]; T[n_, k_] := B[n, n - k + 1];
%t A350262 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A350262 Cf. A350256, A350257, A350258, A350259, A350260, A350261, A350263.
%Y A350262 Cf. A000587, A009235.
%K A350262 sign,tabl
%O A350262 0,4
%A A350262 _Peter Luschny_, Dec 22 2021