A271451 Triangle read by rows of coefficients of polynomials Q_n(x) = 2^(-n)*((x + sqrt(x*(x + 6) - 3) + 1)^n - (x - sqrt(x*(x + 6) - 3) + 1)^n)/sqrt(x*(x + 6) - 3).

1, 1, 1, 0, 3, 1, -1, 3, 5, 1, -1, -1, 10, 7, 1, 0, -6, 7, 21, 9, 1, 1, -6, -10, 31, 36, 11, 1, 1, 1, -29, 7, 79, 55, 13, 1, 0, 9, -24, -63, 81, 159, 78, 15, 1, -1, 9, 15, -123, -54, 264, 279, 105, 17, 1, -1, -1, 57, -69, -321, 132, 624, 447, 136, 19, 1, 0, -12, 50, 126, -459, -507, 741, 1245, 671, 171, 21, 1, 1, -12, -20, 302, -81, -1419, -258, 2163, 2227, 959, 210, 23, 1

Offset

1,5

Comments

The polynomials Q_n(x) have generating function G(x,t) = t/(1 - (x + 1)*t - (x - 1)*t^2) = t + (x + 1)*t^2 + x*(x + 3)*t^3 + (x^3 + 5*x^2 + 3*x - 1)*t^4 + ...

Q_n(x) can be defined by the recurrence relation Q_n(x) = (x + 1)*Q_(n-1)(x) + (x - 1)*Q_(n-2)(x), Q_0(x)=0, Q_1(x)=1.

Discriminants of Q_n(x) gives the sequence: 0, 1, 1, 9, 320, 35600, 10948608, 8664190976, 16836271800320, 77757312009240576, 833309554769920000000, 20346889104219547132493824,...

Q_n(0) = A128834(n).

Q_n(1) = A000079(n-1), n>0.

Q_n(2) = A006190(n).

Q_n(3) = A090017(n).

Q_n(4) = A015536(n).

Q_n(5) = A135032(n).

Q_n(6) = A015562(n).

Q_n(7) = A190560(n).

Q_n(8) = A015583(n).

Q_n(9) = A190957(n).

Q_n(10) = A015603(n).

Links

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened

Ilya Gutkovskiy, Polynomials Q_n(x)

Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Example

Triangle begins:
   1;
   1,  1;
   0,  3,  1;
  -1,  3,  5,  1;
  -1, -1, 10,  7,  1;
...
The first few polynomials are:
Q_0(x) = 0;
Q_1(x) = 1;
Q_2(x) = x + 1;
Q_3(x) = x^2 + 3*x;
Q_4(x) = x^3 + 5*x^2 + 3*x - 1;
Q_5(x) = x^4 + 7*x^3 + 10*x^2 - x - 1,
...

Mathematica

Flatten[Table[CoefficientList[((x + Sqrt[x (x + 6) - 3] + 1)^n - (x - Sqrt[x (x + 6) - 3] + 1)^n)/2^n/Sqrt[x (x + 6) - 3], x], {n, 0, 13}]]

Crossrefs

Cf. A049310.

Sequence in context: A247282 A246685 A218618 * A131248 A116445 A110291

Adjacent sequences: A271448 A271449 A271450 * A271452 A271453 A271454

Keyword

sign,tabl,easy

Author

Ilya Gutkovskiy, Apr 08 2016

Status

approved