A118007 Triangle, diagonals generated from Lucas polynomials.

2, 3, 2, 7, 4, 2, 18, 14, 5, 2, 47, 52, 23, 6, 2, 123, 194, 110, 34, 7, 2, 322, 724, 527, 198, 47, 8, 2, 843, 2702, 2525, 1154, 322, 62, 9, 2, 2207, 10084, 12098, 6726, 2207, 488, 79, 10, 2, 5778, 37634, 57965, 39202, 15127, 3842, 702, 98, 11, 2

Offset: 0

Author

Gary W. Adamson, Apr 09 2006

Keywords

• nonn
• tabl

Comments

Leftmost column = A005248, bisection of Lucas sequence A000032.

Refer to A084534 for a variation of the Lucas polynomials.

Examples

			First few rows of the triangle:
    2;
    3,   2;
    7,   4,   2;
   18,  14,   5,  2;
   47,  52,  23,  6, 2;
  123, 194, 110, 34, 7, 2;
  ...
For example, 4th diagonal from the right (18, 52, 110, ...) = f(x), x=1,2,3, ...: x^3 + 6x^2 + 9x + 2.
(18, 52, 110, ...) = binomial transform of 4th row of A118008: (18, 34, 24, 6).
		

References

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

Crossrefs

Cf. A000032, A005248, A034807, A118008.

Cf. A084534.

Programs

PARI

TLucas(n,k) = binomial(n-k, k) + binomial(n-k-1, k-1) + (n==0); \\ A084534
pol(n) = Pol(vector(n+1, k, TLucas(2*n,k-1)));
T(n,k) = subst(pol(n-k), x, k+1);
trgT(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Aug 12 2022

Formula

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); ...

Diagonals of the triangle are binomial transforms of A118008 rows.

Extensions

More terms from Michel Marcus, Aug 12 2022