Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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