OFFSET
0,8
FORMULA
O.g.f. of column k: 1/(1 - k*x/(1 - 2*k*x/(1 - 3*k*x/(1 - 4*k*x/(1 - 5*k*x/(1 - ...)))))), a continued fraction.
E.g.f. of column k: 1/sqrt(1 - 2*k*x).
A(n,k) = k^n*A001147(n).
EXAMPLE
E.g.f. of column k: A_k(x) = 1 + k*x/1! + 3*k^2*x^2/2! + 15*k^3*x^3/3! + 105*k^4*x^4/4! + 945*k^5*x^5/5! + 10395*k^6*x^6/6! +
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 3, 12, 27, 48, 75, ...
0, 15, 120, 405, 960, 1875, ...
0, 105, 1680, 8505, 26880, 65625, ...
0, 945, 30240, 229635, 967680, 2953125, ...
MATHEMATICA
Table[Function[k, n! SeriesCoefficient[1/Sqrt[1 - 2 k x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-i k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Sep 23 2017
STATUS
approved