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%I #31 Aug 22 2017 21:19:07
%S 1,1,1,1,1,2,1,1,3,5,1,1,5,15,14,1,1,9,61,105,42,1,1,17,297,1385,945,
%T 132,1,1,33,1585,24273,50521,10395,429,1,1,65,8865,485729,3976209,
%U 2702765,135135,1430,1,1,129,50881,10401345,372281761,1145032281,199360981,2027025,4862
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - 2^k*x/(1 - 3^k*x/(1 - 4^k*x/(1 - 5^k*x/(1 - ...)))))).
%F G.f. of column k: 1/(1 - 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: A_k(x) = 1 + x + (2^k + 1)*x^2 + (2^(k+1) + 4^k + 6^k + 1)*x^3 + ...
%e Square array begins:
%e : 1, 1, 1, 1, 1, 1, ...
%e : 1, 1, 1, 1, 1, 1, ...
%e : 2, 3, 5, 9, 17, 33, ...
%e : 5, 15, 61, 297, 1585, 8865, ...
%e : 14, 105, 1385, 24273, 485729, 10401345, ...
%e : 42, 945, 50521, 3976209, 372281761, 38103228225, ...
%t Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-i^k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten
%Y Columns k=0-4 give: A000108, A001147, A000364, A216966, A227887.
%Y Main diagonal gives A291333.
%Y Cf. A000051 (row 2).
%K nonn,tabl
%O 0,6
%A _Ilya Gutkovskiy_, Aug 08 2017