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A291652 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 - ...)))))). 3
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 3, 0, 1, 5, 10, 13, 11, 5, 0, 1, 6, 15, 24, 27, 20, 9, 0, 1, 7, 21, 40, 55, 54, 38, 15, 0, 1, 8, 28, 62, 100, 120, 109, 70, 26, 0, 1, 9, 36, 91, 168, 236, 258, 216, 129, 45, 0, 1, 10, 45, 128, 266, 426, 540, 544, 423, 238, 78, 0, 1, 11, 55, 174, 402, 721, 1035, 1205, 1127, 824, 437, 135, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

FORMULA

G.f. of column k: (1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...)))))))^k, a continued fraction.

EXAMPLE

G.f. of column k: A_k(x) = 1 + k*x + k*(k + 1)*x^2/2 +  k*(k^2 + 3*k + 8)*x^3/6 + k*(k^3 + 6*k^2 + 35*k + 30)*x^4/24 + ...

Square array begins:

1,  1,   1,   1,    1,    1,  ...

0,  1,   2,   3,    4,    5,  ...

0,  1,   3,   6,   10,   15,  ...

0,  2,   6,  13,   24,   40,  ...

0,  3,  11,  27,   55,  100,  ...

0,  5,  20,  54,  120,  236,  ...

MATHEMATICA

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

Table[Function[k, SeriesCoefficient[((Sum[(-1)^i x^(i (i + 1))/Product[(1 - x^m), {m, 1, i}], {i, 0, n}])/(Sum[(-1)^i x^(i^2)/Product[(1 - x^m), {m, 1, i}], {i, 0, n}]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

CROSSREFS

Columns k=0..1 give A000007, A005169.

Rows n=0..3 give A000012, A001477, A000217, A003600 (with a(0)=0).

Main diagonal gives A291653.

Cf. A286509, A286933.

Sequence in context: A279778 A094266 A286335 * A071569 A261835 A286932

Adjacent sequences:  A291649 A291650 A291651 * A291653 A291654 A291655

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Aug 28 2017

STATUS

approved

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Last modified September 22 17:16 EDT 2020. Contains 337291 sequences. (Running on oeis4.)