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A385899 Triangle read by rows: T(n, k, m) = binomial(n, k) * k^n * m^k * (-1)^(n - k) for m = 2.

%I A385899 #20 Aug 03 2025 16:37:56
%S A385899 1,0,2,0,-4,16,0,6,-96,216,0,-8,384,-2592,4096,0,10,-1280,19440,
%T A385899 -81920,100000,0,-12,3840,-116640,983040,-3000000,2985984,0,14,-10752,
%U A385899 612360,-9175040,52500000,-125411328,105413504,0,-16,28672,-2939328,73400320,-700000000,3009871872,-5903156224,4294967296
%N A385899 Triangle read by rows: T(n, k, m) = binomial(n, k) * k^n * m^k * (-1)^(n - k) for m = 2.
%H A385899 Paolo Xausa, <a href="/A385899/b385899.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%e A385899 Triangle begins:
%e A385899   [0] 1;
%e A385899   [1] 0,   2;
%e A385899   [2] 0,  -4,     16;
%e A385899   [3] 0,   6,    -96,      216;
%e A385899   [4] 0,  -8,    384,    -2592,      4096;
%e A385899   [5] 0,  10,  -1280,    19440,    -81920,    100000;
%e A385899   [6] 0, -12,   3840,  -116640,    983040,  -3000000,    2985984;
%e A385899   [7] 0,  14, -10752,   612360,  -9175040,  52500000, -125411328,  105413504;
%p A385899 T := (n, k) -> binomial(n, k) * k^n * 2^k * (-1)^(n - k):
%p A385899 seq(seq(T(n, k), k = 0..n), n = 0..7);
%t A385899 A385899[n_, k_] := If[k == 0, Boole[n == 0], Binomial[n, k]*k^n*2^k*(-1)^(n - k)];
%t A385899 Table[A385899[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Aug 03 2025 *)
%Y A385899 Cf. A000007 (m=0), A258773 (m=1), this sequence (m=2), A062971 (main diagonal), A375540 (row sums), A375541 (row sums of absolute terms).
%K A385899 sign,tabl
%O A385899 0,3
%A A385899 _Peter Luschny_, Aug 02 2025