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A286933 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 A286933 #13 Mar 11 2025 02:55:17
%S A286933 1,1,0,1,1,0,1,2,1,0,1,3,4,2,0,1,4,9,12,3,0,1,5,16,36,32,5,0,1,6,25,
%T A286933 80,135,88,9,0,1,7,36,150,384,513,248,15,0,1,8,49,252,875,1856,1971,
%U A286933 688,26,0,1,9,64,392,1728,5125,9024,7533,1920,45,0,1,10,81,576,3087,11880,30125,43776,28836,5360,78,0
%N A286933 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 A286933 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F A286933 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 A286933 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 A286933 G.f. of column k: A(x) = 1 + k*x + k^2*x^2 + k^2*(k + 1)*x^3 + k^3*(k + 2)*x^4 + k^3*(k^2 + 3*k + 1)*x^5 + ...
%e A286933 Square array begins:
%e A286933   1,  1,   1,    1,     1,     1,  ...
%e A286933   0,  1,   2,    3,     4,     5,  ...
%e A286933   0,  1,   4,    9,    16,    25,  ...
%e A286933   0,  2,  12,   36,    80,   150,  ...
%e A286933   0,  3,  32,  135,   384,   875,  ...
%e A286933   0,  5,  88,  513,  1856,  5125,  ...
%t A286933 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 A286933 Columns k=0..1 give: A000007, A005169.
%Y A286933 Rows n=0..3 give: A000012, A001477, A000290, A011379.
%Y A286933 Main diagonal gives A291274.
%Y A286933 Cf. A286932.
%K A286933 nonn,tabl
%O A286933 0,8
%A A286933 _Ilya Gutkovskiy_, May 16 2017