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A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

%I A232774 #16 Sep 29 2019 11:15:27
%S A232774 1,1,1,1,4,-1,1,11,-5,1,1,26,-16,6,-1,1,57,-42,22,-7,1,1,120,-99,64,
%T A232774 -29,8,-1,1,247,-219,163,-93,37,-9,1,1,502,-466,382,-256,130,-46,10,
%U A232774 -1,1,1013,-968,848,-638,386,-176,56,-11,1,2036,-1981,1816,-1486,1024
%N A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.
%C A232774 Row sums are A000079(n) = 2^n.
%C A232774 Diagonal sums are A024493(n+1) = A130781(n).
%C A232774 Sum_{k=0..n} T(n,k)*x^k = -A003063(n+2), A159964(n), A000012(n), A000079(n), A001045(n+2), A056450(n), (-1)^(n+1)*A232015(n+1) for x = -2, -1, 0, 1, 2, 3, 4 respectively.
%F A232774 G.f.: Sum_{n>=0, k=0..n} T(n,k)*y^k*x^n=(1+2*(y-1)*x)/((1-2*x)*(1+(y-1)*x)).
%F A232774 |T(2*n,n)| = 4^n = A000302(n).
%F A232774 T(n,k) = (-1)^(k-1) * (Sum_{i=0..n-k} (2^(i+1)-1) * binomial(n-i-1,k-1)) for 0 < k <= n and T(n,0) = 1 for n >= 0. - _Werner Schulte_, Mar 22 2019
%e A232774 Triangle begins:
%e A232774   1;
%e A232774   1,    1;
%e A232774   1,    4,   -1;
%e A232774   1,   11,   -5,   1;
%e A232774   1,   26,  -16,   6,   -1;
%e A232774   1,   57,  -42,  22,   -7,   1;
%e A232774   1,  120,  -99,  64,  -29,   8,   -1;
%e A232774   1,  247, -219, 163,  -93,  37,   -9,  1;
%e A232774   1,  502, -466, 382, -256, 130,  -46, 10,  -1;
%e A232774   1, 1013, -968, 848, -638, 386, -176, 56, -11, 1;
%Y A232774 Columns: A000012, A000295, -A002662, A002663, -A002664, A035038, -A035039, A035040, -A035041, A035042.
%Y A232774 Cf. A000079, A055248, A024493, A130781, A000302.
%Y A232774 Cf. also A003063, A159964, A000012, A001045, A056450, A232015.
%K A232774 sign,tabl
%O A232774 0,5
%A A232774 _Philippe Deléham_, Nov 30 2013