A118007 - OEIS

%I #21 Aug 12 2022 09:26:54

%S 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 198,47,8,2,843,2702,2525,1154,322,62,9,2,2207,10084,12098,6726,2207,

%U 488,79,10,2,5778,37634,57965,39202,15127,3842,702,98,11,2

%N Triangle, diagonals generated from Lucas polynomials.

%C Leftmost column = A005248, bisection of Lucas sequence A000032.

%C Refer to A084534 for a variation of the Lucas polynomials.

%D Jay Kappraff, "Beyond Measure, A Guided tour Through Nature, Myth and Number", World Scientific, 2002, p. 485 (Table 22.6b).

%F 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 Diagonals of the triangle are binomial transforms of A118008 rows.

%e First few rows of the triangle:

%e     2;

%e     3,   2;

%e     7,   4,   2;

%e    18,  14,   5,  2;

%e    47,  52,  23,  6, 2;

%e   123, 194, 110, 34, 7, 2;

%e   ...

%e For example, 4th diagonal from the right (18, 52, 110, ...) = f(x), x=1,2,3, ...: x^3 + 6x^2 + 9x + 2.

%e (18, 52, 110, ...) = binomial transform of 4th row of A118008: (18, 34, 24, 6).

%o (PARI) TLucas(n,k) = binomial(n-k, k) + binomial(n-k-1, k-1) + (n==0); \\ A084534

%o pol(n) = Pol(vector(n+1, k, TLucas(2*n,k-1)));

%o T(n,k) = subst(pol(n-k), x, k+1);

%o trgT(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Aug 12 2022

%Y Cf. A000032, A005248, A034807, A118008.

%Y Cf. A084534.

%K nonn,tabl

%O 0,1

%A _Gary W. Adamson_, Apr 09 2006

%E More terms from _Michel Marcus_, Aug 12 2022