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A098493
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Triangle T(n,k) read by rows: difference between A098489 and A098490 at triangular rows.
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6
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1, 0, -1, -1, -1, 1, -1, 1, 2, -1, 0, 3, 0, -3, 1, 1, 2, -5, -2, 4, -1, 1, -2, -7, 6, 5, -5, 1, 0, -5, 0, 15, -5, -9, 6, -1, -1, -3, 12, 9, -25, 1, 14, -7, 1, -1, 3, 15, -18, -29, 35, 7, -20, 8, -1, 0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1, 1, 4, -22, -24, 85, 14, -112, 42
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OFFSET
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0,9
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COMMENTS
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Also, coefficients of polynomials that have values in A098495 and A094954.
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LINKS
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FORMULA
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Recurrence: T(n, k) = T(n-1, k)-T(n-1, k-1)-T(n-2, k); T(n, k)=0 for n<0, k>n, k<0; T(n, n)=(-1)^n; T(n, n-1)=(-1)^n*(1-n).
Riordan array ( (1 - x)/(1 - x + x^2), -x/(1 - x + x^2) ).
T(n,k) = (-1)^k * the (n,k)-th entry of Q^(-1)*P = Sum_{j = k..n} (-1)^(k+binomial(n-j+1,2))*binomial(floor((1/2)*n+(1/2)*j),j)* binomial(j,k), where P denotes Pascal's triangle A007318 and Q denotes triangle A061554 (formed from P by sorting the rows into descending order). (End)
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EXAMPLE
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Triangle begins:
1;
0, -1;
-1, -1, 1;
-1, 1, 2, -1;
0, 3, 0, -3, 1;
...
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MAPLE
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add((-1)^(k+binomial(n-j+1, 2))*binomial(floor((1/2)*n+(1/2)*j), j)* binomial(j, k), j = k..n);
end proc:
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PROG
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(PARI) T(n, k)=if(k>n||k<0||n<0, 0, if(k>=n-1, (-1)^n*if(k==n, 1, -k), if(n==1, 0, if(k==0, T(n-1, 0)-T(n-2, 0), T(n-1, k)-T(n-2, k)-T(n-1, k-1)))))
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CROSSREFS
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Cf. A098494 (diagonal polynomials).
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KEYWORD
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AUTHOR
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STATUS
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approved
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