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 - ...)))))).

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

Links

Table of n, a(n) for n=0..54.

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

Adjacent sequences: A290566 A290567 A290568 * A290570 A290571 A290572

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Aug 08 2017

Status

approved