OFFSET
0,8
LINKS
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
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)).
EXAMPLE
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 + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, -1, -2, -3, -4, -5, ...
0, 1, 4, 9, 16, 25, ...
0, 0, -4, -18, -48, -100, ...
0, -1, 0, 27, 128, 375, ...
0, 1, 8, -27, -320, -1375, ...
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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Ilya Gutkovskiy, May 16 2017
STATUS
approved