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A286932
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Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + k*x/(1 + k*x^2/(1 + k*x^3/(1 + k*x^4/(1 + k*x^5/(1 + ...)))))).
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2
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1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 4, 0, 0, 1, -4, 9, -4, -1, 0, 1, -5, 16, -18, 0, 1, 0, 1, -6, 25, -48, 27, 8, -1, 0, 1, -7, 36, -100, 128, -27, -24, 1, 0, 1, -8, 49, -180, 375, -320, -27, 48, 0, 0, 1, -9, 64, -294, 864, -1375, 704, 243, -64, -1, 0, 1, -10, 81, -448, 1715, -4104, 4875, -1280, -810, 48, 2, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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G.f. of column k: 1/(1 + k*x/(1 + k*x^2/(1 + k*x^3/(1 + k*x^4/(1 + k*x^5/(1 + ...)))))), a continued fraction.
G.f. of column k (for k > 0): (Sum_{j>=0} k^j*x^(j*(j+1))/Product(i=1..j} (1 - x^i)) / (Sum_{j>=0} k^j*x^(j^2)/Product(i=1..j} (1 - x^i)).
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EXAMPLE
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G.f. of column k: A(x) = 1 - k*x + k^2*x^2 - (k - 1)*k^2*x^3 + (k - 2)*k^3*x^4 - k^3*(k^2 - 3*k + 1)*x^5 + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, -1, -2, -3, -4, -5, ...
0, 1, 4, 9, 16, 25, ...
0, 0, -4, -18, -48, -100, ...
0, -1, 0, 27, 128, 375, ...
0, 1, 8, -27, -320, -1375, ...
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MATHEMATICA
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Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[k x^i, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
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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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