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A137372 Triangle read by rows: coefficients of Fermat-Lucas polynomials.

%I A137372 #13 Feb 16 2025 08:33:07
%S A137372 2,0,3,-4,0,9,0,-18,0,27,8,0,-72,0,81,0,60,0,-270,0,243,-16,0,324,0,
%T A137372 -972,0,729,0,-168,0,1512,0,-3402,0,2187,32,0,-1152,0,6480,0,-11664,0,
%U A137372 6561,0,432,0,-6480,0,26244,0,-39366,0,19683,-64,0,3600,0,-32400,0,102060,0,-131220,0,59049
%N A137372 Triangle read by rows: coefficients of Fermat-Lucas polynomials.
%C A137372 The Fermat-Lucas polynomials F(n) are defined by the recurrence: F(0) = 2, F(1) = 3*y and F(n) = 3*y*F(n - 1) - 2*F(n - 2) for n > 1. - _Andrew Howroyd_, Aug 20 2018
%H A137372 Andrew Howroyd, <a href="/A137372/b137372.txt">Table of n, a(n) for n = 0..1274</a>
%H A137372 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fermat-LucasPolynomial.html">Fermat-Lucas Polynomial</a>.
%F A137372 G.f.: (2 - 3*x*y)/(1 - 3*y*x + 2*x^2). - _Andrew Howroyd_, Aug 21 2018
%e A137372 The first few polynomials are:
%e A137372   2;
%e A137372   3*y;
%e A137372   -4 + 9*y^2;
%e A137372   -18*y + 27*y^3;
%e A137372   ...
%e A137372 Triangle begins:
%e A137372     2;
%e A137372     0, 3;
%e A137372    -4, 0, 9;
%e A137372     0, -18, 0, 27;
%e A137372     8, 0, -72, 0, 81;
%e A137372     0, 60, 0, -270, 0, 243;
%e A137372   -16, 0,324, 0, -972, 0, 729;
%e A137372     0, -168, 0, 1512, 0, -3402, 0, 2187;
%e A137372    32, 0, -1152, 0, 6480, 0, -11664, 0, 6561;
%e A137372     0, 432, 0, -6480, 0, 26244, 0, -39366, 0, 19683;
%e A137372   -64, 0, 3600, 0, -32400, 0, 102060, 0, -131220, 0, 59049;
%e A137372   ...
%t A137372 << Lucas`; Table[ExpandAll[Fermatf[n, x]], {n, 0, 10}]; a = Table[CoefficientList[Fermatf[n, x], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[Fermatf[n, x], x]], {n, 0, 10}]
%o A137372 (PARI) T(n,k)=polcoef(polcoef((2 - 3*x*y)/(1 - 3*y*x + 2*x^2) + O(x*x^n), n, x), k, y);
%o A137372 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Aug 20 2018
%Y A137372 Row sums are A000051.
%K A137372 tabl,sign
%O A137372 0,1
%A A137372 _Roger L. Bagula_, Apr 09 2008
%E A137372 Offset corrected and name clarified by _Andrew Howroyd_, Aug 20 2018