|
%I #10 Dec 10 2016 17:01:50
%S 1,1,1,2,2,0,3,5,1,-1,5,10,4,-2,-1,8,20,12,-4,-4,0,13,38,31,-4,-13,-2,
%T 1,21,71,73,3,-33,-11,3,1,34,130,162,34,-74,-42,6,6,0,55,235,344,128,
%U -146,-130,0,24,3,-1
%N Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
%C Row-reversed variant of A123585. Row sums: 2^n.
%F G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
%F T(n,0) = A000045(n+1).
%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if n<k or if k<0.
%F Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
%F Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.
%e Triangle begins:
%e 1
%e 1, 1
%e 2, 2, 0
%e 3, 5, 1, -1
%e 5, 10, 4, -2, -1
%e 8, 20, 12, -4, -4, 0
%e 13, 38, 31, -4, -13, -2, 1
%e 21, 71, 73, 3, -33, -11, 3, 1
%e 34, 130, 162, 34, -74, -42, 6, 6, 0
%e 55, 235, 344, 128, -146, -130, 0, 24, 3, -1
%Y Cf. Columns: A000045, A001629, A129707.
%Y Diagonals: A010892, A099254, Antidiagonal sums: A158943.
%K sign,tabl
%O 0,4
%A _Philippe Deléham_, Dec 06 2011
|
