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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)).

%I A152063 #49 Jan 05 2025 19:51:39
%S A152063 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,
%T A152063 55,1,14,74,183,210,89,1,15,86,234,300,144,1,17,115,394,717,654,233,6,
%U A152063 18,130,480,951,954,377,1,20,165,725,1825,2622,1985,610,1,21,183,855
%N 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)).
%C A152063 The triangle A125076 is formed by reading upward sloping diagonals. - _Gary W. Adamson_, Nov 26 2008
%C A152063 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
%H A152063 James P. Bradshaw, Philipp Lampe, and Dusan Ziga, <a href="https://arxiv.org/abs/1910.11823">Snake graphs and their characteristic polynomials</a>, arXiv:1910.11823 [math.CO], 2019. See 4.7 p. 16.
%H A152063 N. D. Cahill and D. A. Narayan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-3/quartcahill03_2004.pdf">Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants</a>, Fibonacci Quarterly, 42(3):216-221, 2004.
%H A152063 M. X. He, D. Simon and P. E. Ricci, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/35-2/he.pdf">Dynamics of the zeros of Fibonacci polynomials</a>, Fibonacci Quarterly, 35(2):160-168, 1997.
%H A152063 V. E. Hoggatt and C. T. Long, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/12-2/hoggatt1.pdf">Divisibility Properties of Generalized Fibonacci Polynomials</a>, Fibonacci Quarterly, 12:113-120, 1974.
%F A152063 Recurrence (as monic polynomials) P(n+4) = (1 + 3*q)*P(n+2) - q^2*P(n). - _F. Chapoton_, May 27 2024
%F A152063 As monic polynomials, these are the numerators of the polynomials from A011973 evaluated at 1/(1+q). - _F. Chapoton_, May 28 2024
%e A152063 First few rows of the triangle are:
%e A152063 1;
%e A152063 1;
%e A152063 1, 2;
%e A152063 1, 3;
%e A152063 1, 5, 5;
%e A152063 1, 6, 8;
%e A152063 1, 8, 19, 13;
%e A152063 1, 9, 25, 21;
%e A152063 1, 11, 42, 65, 34;
%e A152063 1, 12, 51, 90, 55;
%e A152063 1, 14, 74, 183, 210, 89;
%e A152063 1, 15, 86, 234, 300, 144;
%e A152063 1, 17, 115, 394, 717, 654, 233;
%e A152063 1, 18, 130, 480, 951, 954, 377;
%e A152063 1, 20, 165, 725, 1825, 2622, 1985, 610;
%e A152063 1, 21, 183, 855, 2305, 3573, 2939, 987;
%e A152063 ...
%e A152063 By row, alternate signs (+,-,+,-,...) with descending exponents. Rows with n terms have exponents (n-1), (n-2), (n-3),...;
%e A152063 Example: There are two rows with 4 terms corresponding to the polynomials
%e A152063 x^3 - 8x^2 + 19x - 13 (roots associated with the heptagon); and
%e A152063 x^3 - 9x^2 + 25x - 21 (roots associated with the 9-gon (nonagon)).
%p A152063 P := proc(n) option remember; if n < 5 then return
%p A152063 ifelse(n < 3, 1, ifelse(n = 3, 1 + 2*q, 1 + 3*q)) fi;
%p A152063 (1 + 3*q)*P(n - 2) - q^2*P(n - 4) end:
%p A152063 T := n -> local k; seq(coeff(P(n), q, k), k = 0..(n-1)/2):
%p A152063 for n from 1 to 12 do T(n) od;  # (after _F. Chapoton_)  _Peter Luschny_, May 27 2024
%p A152063 # Alternative:
%p A152063 P := n -> local k; add(binomial(n-k,k)*(1+x)^(floor(n/2)-k)*x^k, k=0..floor(n/2)):
%p A152063 T := n -> local k; seq(coeff(P(n), x, k), k = 0..n/2):
%p A152063 for n from 0 to 12 do T(n) od; # (after _F. Chapoton_) _Peter Luschny_, May 28 2024
%Y A152063 Cf. A000045, A002530 (row sums), A125076, A126124, A123965, A011973.
%K A152063 nonn,tabf
%O A152063 1,4
%A A152063 _Gary W. Adamson_ and _Roger L. Bagula_, Nov 22 2008