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A286932 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 + ...)))))).

%I A286932 #15 Mar 11 2025 02:55:28
%S A286932 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 A286932 -6,25,-48,27,8,-1,0,1,-7,36,-100,128,-27,-24,1,0,1,-8,49,-180,375,
%U A286932 -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 A286932 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 A286932 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F A286932 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 A286932 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 A286932 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 A286932 Square array begins:
%e A286932   1,  1,  1,   1,    1,     1,  ...
%e A286932   0, -1, -2,  -3,   -4,    -5,  ...
%e A286932   0,  1,  4,   9,   16,    25,  ...
%e A286932   0,  0, -4, -18,  -48,  -100,  ...
%e A286932   0, -1,  0,  27,  128,   375,  ...
%e A286932   0,  1,  8, -27, -320, -1375,  ...
%t A286932 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 A286932 Columns k=0..1 give: A000007, A007325.
%Y A286932 Rows n=0..3 give: A000012, A001489, A000290, A045991 (gives absolute value).
%Y A286932 Main diagonal gives A291335.
%Y A286932 Cf. A286509.
%K A286932 sign,tabl
%O A286932 0,8
%A A286932 _Ilya Gutkovskiy_, May 16 2017