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%I #6 Sep 23 2017 10:57:05
%S 1,1,0,1,1,0,1,2,3,0,1,3,12,15,0,1,4,27,120,105,0,1,5,48,405,1680,945,
%T 0,1,6,75,960,8505,30240,10395,0,1,7,108,1875,26880,229635,665280,
%U 135135,0,1,8,147,3240,65625,967680,7577955,17297280,2027025,0,1,9,192,5145,136080,2953125,42577920,295540245,518918400,34459425,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/sqrt(1 - 2*k*x).
%F 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.
%F E.g.f. of column k: 1/sqrt(1 - 2*k*x).
%F A(n,k) = k^n*A001147(n).
%e 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! +
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 3, 12, 27, 48, 75, ...
%e 0, 15, 120, 405, 960, 1875, ...
%e 0, 105, 1680, 8505, 26880, 65625, ...
%e 0, 945, 30240, 229635, 967680, 2953125, ...
%t Table[Function[k, n! SeriesCoefficient[1/Sqrt[1 - 2 k x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t 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
%Y Columns k=0..4 give A000007, A001147, A001813, A011781, A144828.
%Y Rows n=0.2 give A000012, A001477, A033428.
%Y Main diagonal gives A292784.
%Y Cf. A131182.
%K nonn,tabl
%O 0,8
%A _Ilya Gutkovskiy_, Sep 23 2017
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