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

0,6

Comments

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.

Interleaving of A000384, A001105, A056220 and A014107 (starting at the second term).

Main diagonal of T is in A001057, row sums are in A131805.

Links

Table of n, a(n) for n=0..61.

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).

Example

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 ]

Prog

(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] ];
(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, ", "))}

Crossrefs

Cf. A129819, A000384 (n*(2*n-1)), A001105 (2*n^2), A056220 (2*n^2-1), A014107 (n*(2*n-3)), A001057, A131805.

Sequence in context: A020825 A259992 A110422 * A307519 A254198 A246466

Adjacent sequences: A131801 A131802 A131803 * A131805 A131806 A131807

Keyword

sign

Author

Klaus Brockhaus, Jul 18 2007

Status

approved