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%I #3 Sep 08 2022 08:45:31
%S 0,0,-1,-1,1,2,1,2,6,8,7,9,15,18,17,20,28,32,31,35,45,50,49,54,66,72,
%T 71,77,91,98,97,104,120,128,127,135,153,162,161,170,190,200,199,209,
%U 231,242,241,252,276,288,287,299,325,338,337,350,378,392,391,405,435,450
%N Antidiagonal sums of triangular array T: T(j,k) = -(k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.
%C T is obtained by replacing the values of the second, fourth, sixth, ... column of the triangular array defined in A129819 by the corresponding negative values.
%C Interleaving of A000384, A001105, A056220 and A014107 (starting at the second term).
%C Main diagonal of T is in A001057, row sums are in A131805.
%F a(0) = 0, a(1) = 0, a(2) = -1, a(3) = -1, a(4) = 1, a(5) = 2, a(6) = 1; for n > 6, a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7);
%F G.f.: x^2*(-1+2*x-x^2+x^3)/((1-x)^3*(1+x^2)^2).
%e First seven rows of T are
%e [ 0 ],
%e [ 0, -1 ],
%e [ 0, -1, 2 ],
%e [ 0, -1, 3, -2 ],
%e [ 0, -1, 4, -2, 3 ],
%e [ 0, -1, 5, -2, 4, -3 ],
%e [ 0, -1, 6, -2, 5, -3, 4 ]
%o (Magma) m:=62; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:=-k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ];
%o (PARI) {for(n=0, 61, r=n%4; k=(n-r)/4; a=if(r==0, k*(2*k-1), if(r==1, 2*k^2, if(r==2, 2*k^2-1, k*(2*k+1)-1))); print1(a, ","))}
%Y Cf. A129819, A000384 (n*(2*n-1)), A001105 (2*n^2), A056220 (2*n^2-1), A014107 (n*(2*n-3)), A001057, A131805.
%K sign
%O 0,6
%A _Klaus Brockhaus_, Jul 18 2007
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