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A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

%I A123585 #18 Oct 17 2017 09:29:14
%S A123585 1,1,1,0,2,2,-1,1,5,3,-1,-2,4,10,5,0,-4,-4,12,20,8,1,-2,-13,-4,31,38,
%T A123585 13,1,3,-11,-33,3,73,71,21,0,6,6,-42,-74,34,162,130,34,-1,3,24,0,-130,
%U A123585 -146,128,344,235,55,-1,-4,21,72,-50,-352
%N A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
%H A123585 G. C. Greubel, <a href="/A123585/b123585.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A123585 Sum_{k,0<=k<=n} T(n,k) = 2^n = A000079(n).
%F A123585 T(n,0) = A010892(n).
%F A123585 T(n,n) = Fibonacci(n+1) = A000045(n+1).
%F A123585 T(n+1,1) = A099254(n).
%F A123585 T(n+1,n) = A001629(n+2).
%F A123585 Sum_{k, 0<=k<=[n/2]} T(n-k,k) = A003269(n).
%F A123585 T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k), n>0.
%F A123585 Sum_{k, 0<=k<=n} x^k*T(n,k) = (-1)^n*A003683(n+1), (-1)^n*A006130(n), A000007(n), A010892(n), A000079(n), A030195(n+1) for x=-3, -2, -1, 0, 1, 2 respectively . - _Philippe Deléham_, Dec 01 2006
%F A123585 T(n+2,n) = A129707(n+1).- _Philippe Deléham_, Dec 18 2011
%F A123585 G.f.: 1/(1-(1+y)*x+(1-y^2)*x^2). - _Philippe Deléham_, Dec 18 2011
%e A123585 Triangle begins:
%e A123585 1;
%e A123585 1, 1;
%e A123585 0, 2, 2;
%e A123585 -1, 1, 5, 3;
%e A123585 -1, -2, 4, 10, 5;
%e A123585 0, -4, -4, 12, 20, 8;
%e A123585 1, -2, -13, -4, 31, 38, 13;
%e A123585 1, 3, -11, -33, 3, 73, 71, 21;
%e A123585 0, 6, 6, -42, -74, 34, 162, 130, 34;
%t A123585 CoefficientList[CoefficientList[Series[1/(1 - (1 + y)*x + (1 - y^2)*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* _G. C. Greubel_, Oct 16 2017 *)
%Y A123585 Cf. A010892, A099254, A000045, A000079 (row sums), A001629, A129707.
%K A123585 sign,tabl
%O A123585 0,5
%A A123585 _Philippe Deléham_, Nov 13 2006