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A290569
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 - ...)))))).
5
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, 132, 1, 1, 33, 1585, 24273, 50521, 10395, 429, 1, 1, 65, 8865, 485729, 3976209, 2702765, 135135, 1430, 1, 1, 129, 50881, 10401345, 372281761, 1145032281, 199360981, 2027025, 4862
OFFSET
0,6
FORMULA
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.
EXAMPLE
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 + ...
Square array begins:
: 1, 1, 1, 1, 1, 1, ...
: 1, 1, 1, 1, 1, 1, ...
: 2, 3, 5, 9, 17, 33, ...
: 5, 15, 61, 297, 1585, 8865, ...
: 14, 105, 1385, 24273, 485729, 10401345, ...
: 42, 945, 50521, 3976209, 372281761, 38103228225, ...
MATHEMATICA
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
CROSSREFS
Columns k=0-4 give: A000108, A001147, A000364, A216966, A227887.
Main diagonal gives A291333.
Cf. A000051 (row 2).
Sequence in context: A114163 A189435 A279636 * A230698 A090234 A286380
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Aug 08 2017
STATUS
approved