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Search: id:A131804
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| A131804 |
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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. |
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+0
2
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| 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
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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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.
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FORMULA
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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).
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EXAMPLE
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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 ]
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PROGRAM
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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] ];
(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, ", "))}
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CROSSREFS
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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: A070236 A020825 A110422 this_sequence A032085 A032163 A038078
Adjacent sequences: A131801 A131802 A131803 this_sequence A131805 A131806 A131807
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KEYWORD
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sign
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AUTHOR
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Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 18 2007
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