A286932 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + k*x/(1 + k*x^2/(1 + k*x^3/(1 + k*x^4/(1 + k*x^5/(1 + ...)))))).

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 4, 0, 0, 1, -4, 9, -4, -1, 0, 1, -5, 16, -18, 0, 1, 0, 1, -6, 25, -48, 27, 8, -1, 0, 1, -7, 36, -100, 128, -27, -24, 1, 0, 1, -8, 49, -180, 375, -320, -27, 48, 0, 0, 1, -9, 64, -294, 864, -1375, 704, 243, -64, -1, 0, 1, -10, 81, -448, 1715, -4104, 4875, -1280, -810, 48, 2, 0

Offset

0,8

Links

Table of n, a(n) for n=0..77.

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

Formula

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

G.f. of column k (for k > 0): (Sum_{j>=0} k^j*x^(j*(j+1))/Product(i=1..j} (1 - x^i)) / (Sum_{j>=0} k^j*x^(j^2)/Product(i=1..j} (1 - x^i)).

Example

G.f. of column k: A(x) = 1 - k*x + k^2*x^2 - (k - 1)*k^2*x^3 + (k - 2)*k^3*x^4 - k^3*(k^2 - 3*k + 1)*x^5 + ...
Square array begins:
1,  1,  1,   1,    1,     1,  ...
0, -1, -2,  -3,   -4,    -5,  ...
0,  1,  4,   9,   16,    25,  ...
0,  0, -4, -18,  -48,  -100,  ...
0, -1,  0,  27,  128,   375,  ...
0,  1,  8, -27, -320, -1375,  ...

Mathematica

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[k x^i, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0-1 give: A000007, A007325.

Rows n=0-3 give: A000012, A001489, A000290, A045991 (gives absolute value).

Main diagonal gives A291335.

Cf. A286509.

Sequence in context: A378321 A261835 A353436 * A350364 A358575 A259475

Adjacent sequences: A286929 A286930 A286931 * A286933 A286934 A286935

Keyword

sign,tabl

Author

Ilya Gutkovskiy, May 16 2017

Status

approved