%I #9 Feb 16 2025 08:33:45

%S 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 80,135,88,9,0,1,7,36,150,384,513,248,15,0,1,8,49,252,875,1856,1971,

%U 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 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="https://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^2*(k + 1)*x^3 + k^3*(k + 2)*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, 2, 12, 36, 80, 150, ...

%e 0, 3, 32, 135, 384, 875, ...

%e 0, 5, 88, 513, 1856, 5125, ...

%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, A005169.

%Y Rows n=0-3 give: A000012, A001477, A000290, A011379.

%Y Main diagonal gives A291274.

%Y Cf. A286932.

%K nonn,tabl

%O 0,8

%A _Ilya Gutkovskiy_, May 16 2017