1,4

Left border of the triangle = Fibonacci numbers, right border = factorials. Companion triangle A117937 is generated from Lucas polynomials, using analogous operations.

Inverse binomial transforms of A073133 columns. Such columns are f(x), Fibonacci polynomials.

First few columns of A073133 are: (1, 1, 1,...); (1, 2, 3,...); (2, 5, 10, 17,...); (3, 12, 33, 72,...). As sequences, these are f(x), Fibonacci polynomials: (1); (x); (x^2 + 1); (x^3 + 2x); (x^4 + 3x^2 + 1); (x^5 + 4x^3 + 3x);...For example, f(x), x = 1,2,3...using (x^4 + 3x^2 + 1) generates Column 5 of A073133: (5, 29, 109, 305...).

Inverse binomial transforms of the foregoing columns generates the triangle rows:

1;

1, 1;

2, 3, 2;

3, 9, 12, 6;

5, 24, 56, 60, 24;

8, 62, 228, 414, 360, 120;

...

Cf. A073133, A117937, A117938.

Sequence in context: A076731 A085216 A102310 this_sequence A078331 A093868 A077942

Adjacent sequences: A117933 A117934 A117935 this_sequence A117937 A117938 A117939

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 03 2006