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A137374 Triangular array of the coefficients of the Jacobsthal-Lucas polynomials ordered by descending powers, read by rows.

%I A137374 #28 Feb 16 2025 08:33:07
%S A137374 2,1,4,1,6,1,8,8,1,20,10,1,16,36,12,1,56,56,14,1,32,128,80,16,1,144,
%T A137374 240,108,18,1,64,400,400,140,20,1,352,880,616,176,22,1,128,1152,1680,
%U A137374 896,216,24,1,832,2912,2912,1248,260,26,1,256,3136,6272,4704,1680,308,28,1
%N A137374 Triangular array of the coefficients of the Jacobsthal-Lucas polynomials ordered by descending powers, read by rows.
%C A137374 The even rows which start with 4, 8, 16 ... appear to be the absolute values of the Riordan array A128414. - _Georg Fischer_, Feb 25 2020
%H A137374 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Jacobsthal-LucasPolynomial.html">Jacobsthal-Lucas Polynomial</a>.
%F A137374 Let p(n, x) = 2*x*p(n-2, x) + p(n-1, x) with p(0, x) = 2 and p(1, x) = 1. The coefficients of the polynomial p(n, x), listed in reverse order, give row n. - _Jinyuan Wang_, Feb 25 2020
%e A137374 The triangle starts:
%e A137374     2;
%e A137374     1;
%e A137374     4,   1;
%e A137374     6,   1;
%e A137374     8,   8,   1;
%e A137374    20,  10,   1;
%e A137374    16,  36,  12,   1;
%e A137374    56,  56,  14,   1;
%e A137374    32, 128,  80,  16,  1;
%e A137374   144, 240, 108,  18,  1;
%e A137374    64, 400, 400, 140, 20, 1;
%e A137374   352, 880, 616, 176, 22, 1;
%e A137374   ...
%p A137374 b:= proc(n) option remember;
%p A137374       `if`(n<2, 2-n, b(n-1)+2*expand(x*b(n-2)))
%p A137374     end:
%p A137374 T:= n-> (p-> (d-> seq(coeff(p, x, d-i), i=0..d))(degree(p)))(b(n)):
%p A137374 seq(T(n), n=0..20);  # _Alois P. Heinz_, Feb 25 2020
%t A137374 f[0] = 2; f[1] = 1; f[n_] := 2 x f[n - 2] + f[n - 1];
%t A137374 Table[Reverse[CoefficientList[f[n], x]], {n, 0, 14}] // Flatten (* _Jinyuan Wang_, Feb 25 2020 *)
%Y A137374 Row sums give A014551.
%Y A137374 Cf. A034807.
%K A137374 nonn,tabf
%O A137374 0,1
%A A137374 _Roger L. Bagula_, Apr 09 2008
%E A137374 Offset set to 0 by _Peter Luschny_, Feb 25 2020