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A118007 Triangle, diagonals generated from Lucas polynomials.

%I A118007 #21 Aug 12 2022 09:26:54
%S A118007 2,3,2,7,4,2,18,14,5,2,47,52,23,6,2,123,194,110,34,7,2,322,724,527,
%T A118007 198,47,8,2,843,2702,2525,1154,322,62,9,2,2207,10084,12098,6726,2207,
%U A118007 488,79,10,2,5778,37634,57965,39202,15127,3842,702,98,11,2
%N A118007 Triangle, diagonals generated from Lucas polynomials.
%C A118007 Leftmost column = A005248, bisection of Lucas sequence A000032.
%C A118007 Refer to A084534 for a variation of the Lucas polynomials.
%D A118007 Jay Kappraff, "Beyond Measure, A Guided tour Through Nature, Myth and Number", World Scientific, 2002, p. 485 (Table 22.6b).
%F A118007 Diagonals are sequences as f(x), x=1,2,3; Lucas polynomials in the format: (2); (x + 2); (x^2 + 4x + 2); (x^3 + 6x^2 + 9x + 2); (x^4 + 8x^3 + 20x^2 + 16x + 2); (x^5 + 10x^4 + 35x^3 + 50x^2 + 25x + 2); ...
%F A118007 Diagonals of the triangle are binomial transforms of A118008 rows.
%e A118007 First few rows of the triangle:
%e A118007     2;
%e A118007     3,   2;
%e A118007     7,   4,   2;
%e A118007    18,  14,   5,  2;
%e A118007    47,  52,  23,  6, 2;
%e A118007   123, 194, 110, 34, 7, 2;
%e A118007   ...
%e A118007 For example, 4th diagonal from the right (18, 52, 110, ...) = f(x), x=1,2,3, ...: x^3 + 6x^2 + 9x + 2.
%e A118007 (18, 52, 110, ...) = binomial transform of 4th row of A118008: (18, 34, 24, 6).
%o A118007 (PARI) TLucas(n,k) = binomial(n-k, k) + binomial(n-k-1, k-1) + (n==0); \\ A084534
%o A118007 pol(n) = Pol(vector(n+1, k, TLucas(2*n,k-1)));
%o A118007 T(n,k) = subst(pol(n-k), x, k+1);
%o A118007 trgT(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Aug 12 2022
%Y A118007 Cf. A000032, A005248, A034807, A118008.
%Y A118007 Cf. A084534.
%K A118007 nonn,tabl
%O A118007 0,1
%A A118007 _Gary W. Adamson_, Apr 09 2006
%E A118007 More terms from _Michel Marcus_, Aug 12 2022