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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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%I #8 Aug 23 2017 09:50:23

%S 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,

%T -6,25,-48,27,8,-1,0,1,-7,36,-100,128,-27,-24,1,0,1,-8,49,-180,375,

%U -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

%N 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 + ...)))))).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>

%F 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.

%F 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)).

%e 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 + ...

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, -1, -2, -3, -4, -5, ...

%e 0, 1, 4, 9, 16, 25, ...

%e 0, 0, -4, -18, -48, -100, ...

%e 0, -1, 0, 27, 128, 375, ...

%e 0, 1, 8, -27, -320, -1375, ...

%t 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

%Y Columns k=0-1 give: A000007, A007325.

%Y Rows n=0-3 give: A000012, A001489, A000290, A045991 (gives absolute value).

%Y Main diagonal gives A291335.

%Y Cf. A286509.

%K sign,tabl

%O 0,8

%A _Ilya Gutkovskiy_, May 16 2017