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