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A286509 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))).

%I A286509 #17 Feb 16 2025 08:33:45
%S A286509 1,1,0,1,-1,0,1,-2,1,0,1,-3,3,0,0,1,-4,6,-2,-1,0,1,-5,10,-7,-1,1,0,1,
%T A286509 -6,15,-16,3,4,-1,0,1,-7,21,-30,15,6,-6,1,0,1,-8,28,-50,40,0,-17,6,0,
%U A286509 0,1,-9,36,-77,84,-26,-30,24,-3,-1,0,1,-10,45,-112,154,-90,-30,64,-21,-2,2,0,1,-11,55,-156,258,-217,15,125,-81,6,9,-3,0
%N A286509 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))).
%H A286509 Seiichi Manyama, <a href="/A286509/b286509.txt">Antidiagonals n = 0..139, flattened</a>
%H A286509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F A286509 G.f. of column k: Product_{j>=1} ((1 - x^(5*j-1))*(1 - x^(5*j-4)) / ((1 - x^(5*j-2))*(1 - x^(5*j-3))))^k.
%e A286509 Square array begins:
%e A286509 1,  1,  1,  1,   1,   1,  ...
%e A286509 0, -1, -2, -3,  -4,  -5,  ...
%e A286509 0,  1,  3,  6,  10,  15,  ...
%e A286509 0,  0, -2, -7, -16, -30,  ...
%e A286509 0, -1, -1,  3,  15,  40,  ...
%e A286509 0,  1,  4,  6,   0, -26,  ...
%t A286509 Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[x^i, 1, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
%t A286509 Table[Function[k, SeriesCoefficient[Product[(1 - x^(5 i - 1)) (1 - x^(5 i - 4))/((1 - x^(5 i - 2)) (1 - x^(5 i - 3))), {i, n}]^k, {x, 0, n}]][j - n], {j, 0, 12},{n, 0, j}] // Flatten
%Y A286509 Columns k=0-5 give: A000007, A007325, A055101, A055102, A055103, A078905 (with offset 0).
%Y A286509 Rows n=0-2 give: A000012, A001489, A000217.
%Y A286509 Main diagonal gives A291651.
%Y A286509 Antidiagonal sums give A302015.
%K A286509 sign,tabl
%O A286509 0,8
%A A286509 _Ilya Gutkovskiy_, May 10 2017