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

%I A356117 #14 Aug 23 2022 13:37:33
%S A356117 0,1,3,3,14,45,7,45,186,630,15,124,630,2540,8925,31,315,1905,8925,
%T A356117 35770,128898,63,762,5355,28616,128898,515844,1891890,127,1785,14308,
%U A356117 85932,429870,1891890,7568484,28113228,255,4088,36828,245640,1351350,6487272,28113228,112456344,421717725
%N A356117 T(n, k) = [x^k] (1/2 - x)^(-n) - (1 - x)^(-n).
%F A356117 T(n, k) = (2^(n+k) - 1) * binomial(n+k-1, k). - _John Keith_, Aug 23 2022
%e A356117 Triangle T(n, k) starts:
%e A356117 [0]   0;
%e A356117 [1]   1,    3;
%e A356117 [2]   3,   14,    45;
%e A356117 [3]   7,   45,   186,    630;
%e A356117 [4]  15,  124,   630,   2540,    8925;
%e A356117 [5]  31,  315,  1905,   8925,   35770,  128898;
%e A356117 [6]  63,  762,  5355,  28616,  128898,  515844,  1891890;
%e A356117 [7] 127, 1785, 14308,  85932,  429870, 1891890,  7568484,  28113228;
%e A356117 [8] 255, 4088, 36828, 245640, 1351350, 6487272, 28113228, 112456344, 421717725;
%p A356117 ser := series((1/2 - x)^(-n) - (1 - x)^(-n), x, 20):
%p A356117 seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9);
%t A356117 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 A356117 Cf. A000225 (column 0), A059672 (column 1), A059937 (column 2), A131568 (main diagonal), A134346, A327318.
%K A356117 nonn,tabl
%O A356117 0,3
%A A356117 _Peter Luschny_, Aug 22 2022