A152063 Triangle read by rows. Coefficients of the Fibonacci product polynomials F(n) = Product_{k=1..(n - 1)/2} (1 + 4*cos^2(k*Pi/n)).

1, 1, 1, 2, 1, 3, 1, 5, 5, 1, 6, 8, 1, 8, 19, 13, 1, 9, 25, 21, 1, 11, 42, 65, 34, 1, 12, 51, 90, 55, 1, 14, 74, 183, 210, 89, 1, 15, 86, 234, 300, 144, 1, 17, 115, 394, 717, 654, 233, 6, 18, 130, 480, 951, 954, 377, 1, 20, 165, 725, 1825, 2622, 1985, 610, 1, 21, 183, 855

Offset

1,4

Comments

The triangle A125076 is formed by reading upward sloping diagonals. - Gary W. Adamson, Nov 26 2008

Bisection of the triangle: odd-indexed rows are reversals of the rows of A126124, even-indexed rows are the reversals of the rows of A123965. - Gary W. Adamson, Aug 15 2010

Links

Table of n, a(n) for n=1..68.

James P. Bradshaw, Philipp Lampe, and Dusan Ziga, Snake graphs and their characteristic polynomials, arXiv:1910.11823 [math.CO], 2019. See 4.7 p. 16.

N. D. Cahill and D. A. Narayan, Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants, Fibonacci Quarterly, 42(3):216-221, 2004.

M. X. He, D. Simon and P. E. Ricci, Dynamics of the zeros of Fibonacci polynomials, Fibonacci Quarterly, 35(2):160-168, 1997.

V. E. Hoggatt and C. T. Long, Divisibility Properties of Generalized Fibonacci Polynomials, Fibonacci Quarterly, 12:113-120, 1974.

Formula

Recurrence (as monic polynomials) P(n+4) = (1 + 3*q)*P(n+2) - q^2*P(n). - F. Chapoton, May 27 2024

As monic polynomials, these are the numerators of the polynomials from A011973 evaluated at 1/(1+q). - F. Chapoton, May 28 2024

Example

First few rows of the triangle are:
1;
1;
1, 2;
1, 3;
1, 5, 5;
1, 6, 8;
1, 8, 19, 13;
1, 9, 25, 21;
1, 11, 42, 65, 34;
1, 12, 51, 90, 55;
1, 14, 74, 183, 210, 89;
1, 15, 86, 234, 300, 144;
1, 17, 115, 394, 717, 654, 233;
1, 18, 130, 480, 951, 954, 377;
1, 20, 165, 725, 1825, 2622, 1985, 610;
1, 21, 183, 855, 2305, 3573, 2939, 987;
...
By row, alternate signs (+,-,+,-,...) with descending exponents. Rows with n terms have exponents (n-1), (n-2), (n-3),...;
Example: There are two rows with 4 terms corresponding to the polynomials
x^3 - 8x^2 + 19x - 13 (roots associated with the heptagon); and
x^3 - 9x^2 + 25x - 21 (roots associated with the 9-gon (nonagon)).

Maple

P := proc(n) option remember; if n < 5 then return
ifelse(n < 3, 1, ifelse(n = 3, 1 + 2*q, 1 + 3*q)) fi;
(1 + 3*q)*P(n - 2) - q^2*P(n - 4) end:
T := n -> local k; seq(coeff(P(n), q, k), k = 0..(n-1)/2):
for n from 1 to 12 do T(n) od;  # (after F. Chapoton)  Peter Luschny, May 27 2024
# Alternative:
P := n -> local k; add(binomial(n-k, k)*(1+x)^(floor(n/2)-k)*x^k, k=0..floor(n/2)):
T := n -> local k; seq(coeff(P(n), x, k), k = 0..n/2):
for n from 0 to 12 do T(n) od; # (after F. Chapoton) Peter Luschny, May 28 2024

Crossrefs

Cf. A000045, A002530 (row sums), A125076, A126124, A123965, A011973.

Sequence in context: A078657 A080959 A065548 * A336617 A341865 A347982

Adjacent sequences: A152060 A152061 A152062 * A152064 A152065 A152066

Keyword

nonn,tabf

Author

Gary W. Adamson and Roger L. Bagula, Nov 22 2008

Status

approved