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A104041
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Triangular matrix T, read by rows, such that column k is equal (in absolute value) to row (k-1) in the matrix inverse T^-1 (with extrapolated zeros) for k>0, with T(n,n)=1 (n>=0) and T(n,n-1)=-n (n>=1).
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2
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1, -1, 1, 0, -2, 1, 0, 2, -3, 1, 0, 0, 4, -4, 1, 0, 0, -4, 8, -5, 1, 0, 0, 0, -8, 12, -6, 1, 0, 0, 0, 8, -20, 18, -7, 1, 0, 0, 0, 0, 16, -32, 24, -8, 1, 0, 0, 0, 0, -16, 48, -56, 32, -9, 1, 0, 0, 0, 0, 0, -32, 80, -80, 40, -10, 1, 0, 0, 0, 0, 0, 32, -112, 160, -120, 50, -11, 1, 0, 0, 0, 0, 0, 0, 64, -192, 240, -160, 60, -12, 1
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OFFSET
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0,5
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COMMENTS
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Row sums are: {1,0,-1,0, 1,0,-1,0, ...}. Absolute row sums form A038754. Let A(x,y) be the g.f. of T and B(x,y) be the g.f. of T^-1; then B(x,y)=1+x*y*A(-1/y,-x*y^2) and A(x,y)=(B(-x^2*y,-1/x)-1)/(x*y).
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LINKS
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FORMULA
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G.f.: A(x, y) = (1 - x + x*y)/(1 + 2*x^2*y - x^2*y^2).
T(2*n+1,k) = Sum_{i = k-n-1..n} Stirling2(n,i)*Stirling1(i+2,k+1-n) for 0 <= k <= 2*n+1.
T(2*n,k) = binomial(n,k-n)*(-2)^(2*n-k) for 0 <= k <= 2*n. Cf. A038207. (End)
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EXAMPLE
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Rows of T begin:
1;
-1, 1;
0, -2, 1;
0, 2, -3, 1;
0, 0, 4, -4, 1;
0, 0, -4, 8, -5, 1;
0, 0, 0, -8, 12, -6, 1;
0, 0, 0, 8, -20, 18, -7, 1; ...
The matrix inverse T^-1 equals triangle A104040:
1;
1, 1;
2, 2, 1;
4, 4, 3, 1;
8, 8, 8, 4, 1;
16, 16, 20, 12, 5, 1;
32, 32, 48, 32, 18, 6, 1;
64, 64, 112, 80, 56, 24, 7, 1; ...
The rows of T^-1 equal columns of T in absolute value.
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PROG
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(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff((1-X+X*Y)/(1+2*X^2*Y-X^2*Y^2), n, x), k, y)}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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