A356117 T(n, k) = [x^k] (1/2 - x)^(-n) - (1 - x)^(-n).

0, 1, 3, 3, 14, 45, 7, 45, 186, 630, 15, 124, 630, 2540, 8925, 31, 315, 1905, 8925, 35770, 128898, 63, 762, 5355, 28616, 128898, 515844, 1891890, 127, 1785, 14308, 85932, 429870, 1891890, 7568484, 28113228, 255, 4088, 36828, 245640, 1351350, 6487272, 28113228, 112456344, 421717725

Offset: 0

Author

Peter Luschny, Aug 22 2022

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
[0]   0;
[1]   1,    3;
[2]   3,   14,    45;
[3]   7,   45,   186,    630;
[4]  15,  124,   630,   2540,    8925;
[5]  31,  315,  1905,   8925,   35770,  128898;
[6]  63,  762,  5355,  28616,  128898,  515844,  1891890;
[7] 127, 1785, 14308,  85932,  429870, 1891890,  7568484,  28113228;
[8] 255, 4088, 36828, 245640, 1351350, 6487272, 28113228, 112456344, 421717725;
		

Crossrefs

Cf. A000225 (column 0), A059672 (column 1), A059937 (column 2), A131568 (main diagonal), A134346, A327318.

Programs

Maple

ser := series((1/2 - x)^(-n) - (1 - x)^(-n), x, 20):
seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9);

Mathematica

row[n_] := CoefficientList[Series[(1/2 - x)^(-n) - (1 - x)^(-n), {x, 0, n}], x]; row[0] = {0}; Table[row[n], {n, 0, 8}] // Flatten (* Amiram Eldar, Aug 22 2022 *)

Formula

T(n, k) = (2^(n+k) - 1) * binomial(n+k-1, k). - John Keith, Aug 23 2022