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 - ...)))))).

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

Offset: 0

Author

Ilya Gutkovskiy, May 16 2017

Keywords

• nonn
• tabl

Examples

			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 + ...
Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  0,  1,   2,    3,     4,     5,  ...
  0,  1,   4,    9,    16,    25,  ...
  0,  2,  12,   36,    80,   150,  ...
  0,  3,  32,  135,   384,   875,  ...
  0,  5,  88,  513,  1856,  5125,  ...
		

Links

• Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Crossrefs

Columns k=0..1 give: A000007, A005169.

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

Main diagonal gives A291274.

Cf. A286932.

Programs

Mathematica

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

Formula

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)).