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%I #14 Aug 23 2022 13:37:33
%S 0,1,3,3,14,45,7,45,186,630,15,124,630,2540,8925,31,315,1905,8925,
%T 35770,128898,63,762,5355,28616,128898,515844,1891890,127,1785,14308,
%U 85932,429870,1891890,7568484,28113228,255,4088,36828,245640,1351350,6487272,28113228,112456344,421717725
%N T(n, k) = [x^k] (1/2 - x)^(-n) - (1 - x)^(-n).
%F T(n, k) = (2^(n+k) - 1) * binomial(n+k-1, k). - _John Keith_, Aug 23 2022
%e Triangle T(n, k) starts:
%e [0] 0;
%e [1] 1, 3;
%e [2] 3, 14, 45;
%e [3] 7, 45, 186, 630;
%e [4] 15, 124, 630, 2540, 8925;
%e [5] 31, 315, 1905, 8925, 35770, 128898;
%e [6] 63, 762, 5355, 28616, 128898, 515844, 1891890;
%e [7] 127, 1785, 14308, 85932, 429870, 1891890, 7568484, 28113228;
%e [8] 255, 4088, 36828, 245640, 1351350, 6487272, 28113228, 112456344, 421717725;
%p ser := series((1/2 - x)^(-n) - (1 - x)^(-n), x, 20):
%p seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9);
%t 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 *)
%Y Cf. A000225 (column 0), A059672 (column 1), A059937 (column 2), A131568 (main diagonal), A134346, A327318.
%K nonn,tabl
%O 0,3
%A _Peter Luschny_, Aug 22 2022