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

1, 0, 2, 0, -4, 16, 0, 6, -96, 216, 0, -8, 384, -2592, 4096, 0, 10, -1280, 19440, -81920, 100000, 0, -12, 3840, -116640, 983040, -3000000, 2985984, 0, 14, -10752, 612360, -9175040, 52500000, -125411328, 105413504, 0, -16, 28672, -2939328, 73400320, -700000000, 3009871872, -5903156224, 4294967296

Offset: 0

Author

Peter Luschny, Aug 02 2025

Keywords

• sign
• tabl

Examples

			Triangle begins:
  [0] 1;
  [1] 0,   2;
  [2] 0,  -4,     16;
  [3] 0,   6,    -96,      216;
  [4] 0,  -8,    384,    -2592,      4096;
  [5] 0,  10,  -1280,    19440,    -81920,    100000;
  [6] 0, -12,   3840,  -116640,    983040,  -3000000,    2985984;
  [7] 0,  14, -10752,   612360,  -9175040,  52500000, -125411328,  105413504;
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Crossrefs

Cf. A000007 (m=0), A258773 (m=1), this sequence (m=2), A062971 (main diagonal), A375540 (row sums), A375541 (row sums of absolute terms).

Programs

Maple

T := (n, k) -> binomial(n, k) * k^n * 2^k * (-1)^(n - k):
seq(seq(T(n, k), k = 0..n), n = 0..7);

Mathematica

A385899[n_, k_] := If[k == 0, Boole[n == 0], Binomial[n, k]*k^n*2^k*(-1)^(n - k)];
Table[A385899[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 03 2025 *)