A131804 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.
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, 71, 77, 91, 98, 97, 104, 120, 128, 127, 135, 153, 162, 161, 170, 190, 200, 199, 209, 231, 242, 241, 252, 276, 288, 287, 299, 325, 338, 337, 350, 378, 392, 391, 405, 435, 450
Offset: 0
Keywords
Examples
First seven rows of T are [ 0 ], [ 0, -1 ], [ 0, -1, 2 ], [ 0, -1, 3, -2 ], [ 0, -1, 4, -2, 3 ], [ 0, -1, 5, -2, 4, -3 ], [ 0, -1, 6, -2, 5, -3, 4 ]
Crossrefs
Programs
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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] ];
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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, ","))}
Formula
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);
G.f.: x^2*(-1+2*x-x^2+x^3)/((1-x)^3*(1+x^2)^2).
Comments