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%I #16 Jan 21 2020 21:23:17
%S 1,0,1,0,1,2,0,0,3,4,0,0,1,8,8,0,0,0,5,20,16,0,0,0,1,18,48,32,0,0,0,0,
%T 7,56,112,64,0,0,0,0,1,32,160,256,128,0,0,0,0,0,9,120,432,576,256,0,0,
%U 0,0,0,1,50,400,1120,1280,512
%N A skewed version of triangular array A081277.
%C Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.
%F T(0,0)=T(1,1)=1, T(n,k)=0 if n < k or if k < 0, T(n,k) = T(n-2,k-1) + 2*T(n-1,k-1).
%F Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
%F Sum_{k=0..floor(n/2)} T(n-k,k) = Fibonacci(n-1) = A000045(n-1).
%F Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively. - _Philippe Deléham_, Dec 26 2007
%F Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. - _Philippe Deléham_, Nov 14 2008
%F G.f.: (1-y*x)/(1-2y*x-y*x^2). - _Philippe Deléham_, Dec 04 2011
%F Sum_{k=0..n} T(n,k)^2 = A002002(n) for n > 0. - _Philippe Deléham_, Dec 04 2011
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 0, 3, 4;
%e 0, 0, 1, 8, 8;
%e 0, 0, 0, 5, 20, 16;
%e 0, 0, 0, 1, 18, 48, 32;
%e 0, 0, 0, 0, 7, 56, 112, 64;
%e 0, 0, 0, 0, 1, 32, 160, 256, 128;
%e 0, 0, 0, 0, 0, 9, 120, 432, 576, 256;
%e 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512;
%Y Cf. A025192 (column sums). Diagonals include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
%K nonn,tabl
%O 0,6
%A _Philippe Deléham_, Dec 05 2006
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