A292783 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/sqrt(1 - 2*k*x).

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 12, 15, 0, 1, 4, 27, 120, 105, 0, 1, 5, 48, 405, 1680, 945, 0, 1, 6, 75, 960, 8505, 30240, 10395, 0, 1, 7, 108, 1875, 26880, 229635, 665280, 135135, 0, 1, 8, 147, 3240, 65625, 967680, 7577955, 17297280, 2027025, 0, 1, 9, 192, 5145, 136080, 2953125, 42577920, 295540245, 518918400, 34459425, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

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

E.g.f. of column k: 1/sqrt(1 - 2*k*x).

A(n,k) = k^n*A001147(n).

Example

E.g.f. of column k: A_k(x) = 1 + k*x/1! + 3*k^2*x^2/2! + 15*k^3*x^3/3! + 105*k^4*x^4/4! + 945*k^5*x^5/5! + 10395*k^6*x^6/6! +
Square array begins:
1,    1,      1,       1,       1,        1,  ...
0,    1,      2,       3,       4,        5,  ...
0,    3,     12,      27,      48,       75,  ...
0,   15,    120,     405,     960,     1875,  ...
0,  105,   1680,    8505,   26880,    65625,  ...
0,  945,  30240,  229635,  967680,  2953125,  ...

Mathematica

Table[Function[k, n! SeriesCoefficient[1/Sqrt[1 - 2 k x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-i k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..4 give A000007, A001147, A001813, A011781, A144828.

Rows n=0.2 give A000012, A001477, A033428.

Main diagonal gives A292784.

Cf. A131182.

Sequence in context: A340798 A355427 A122078 * A320354 A285320 A347710

Adjacent sequences: A292780 A292781 A292782 * A292784 A292785 A292786

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Sep 23 2017

Status

approved