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
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).
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
Comments