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A350261 Triangle read by rows. T(n, k) = k^n * BellPolynomial(n, -1/k) for k > 0, if k = 0 then T(n, k) = k^n.

%I A350261 #8 Dec 30 2021 07:23:17
%S A350261 1,0,-1,0,0,-1,0,1,1,-1,0,1,9,19,25,0,-2,23,128,343,674,0,-9,-25,379,
%T A350261 2133,6551,15211,0,-9,-583,-1549,3603,33479,123821,331827,0,50,-3087,
%U A350261 -32600,-112975,-174114,120865,1619108,5987745
%N A350261 Triangle read by rows. T(n, k) = k^n * BellPolynomial(n, -1/k) for k > 0, if k = 0 then T(n, k) = k^n.
%e A350261 Triangle starts:
%e A350261 [0] 1
%e A350261 [1] 0, -1
%e A350261 [2] 0,  0,    -1
%e A350261 [3] 0,  1,     1,     -1
%e A350261 [4] 0,  1,     9,     19,      25
%e A350261 [5] 0, -2,    23,    128,     343,     674
%e A350261 [6] 0, -9,   -25,    379,    2133,    6551,  15211
%e A350261 [7] 0, -9,  -583,  -1549,    3603,   33479, 123821,  331827
%e A350261 [8] 0, 50, -3087, -32600, -112975, -174114, 120865, 1619108, 5987745
%p A350261 A350261 := (n, k) -> ifelse(k = 0, k^n, k^n * BellB(n, -1/k)):
%p A350261 seq(seq(A350261(n, k), k = 0..n), n = 0..8);
%t A350261 T[n_, k_] := If[k == 0, k^n, k^n BellB[n, -1/k]];
%t A350261 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A350261 Cf. A350256, A350257, A350258, A350259, A350260, A350262, A350263.
%Y A350261 Cf. A000587, A318183.
%K A350261 sign,tabl
%O A350261 0,13
%A A350261 _Peter Luschny_, Dec 22 2021