OFFSET
1,4
COMMENTS
Left border of the triangle = Fibonacci numbers, right border = factorials. Companion triangle A117937 is generated from Lucas polynomials, using analogous operations.
Note that binomial transforms are defined from offset 1 here. - R. J. Mathar, Aug 16 2019
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flatten
FORMULA
Inverse binomial transforms of A073133 columns. Such columns are f(x), Fibonacci polynomials.
EXAMPLE
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 + 2*x); (x^4 + 3*x^2 + 1); (x^5 + 4*x^3 + 3*x); ... For example, f(x), x = 1,2,3,... using (x^4 + 3*x^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;
...
MAPLE
A117936 := proc(n, k)
add( A073133(i+1, n)*binomial(k-1, i)*(-1)^(i-k-1), i=0..k-1) ;
end proc:
seq(seq(A117936(n, k), k=1..n), n=1..13) ; # R. J. Mathar, Aug 16 2019
MATHEMATICA
(* A = A073133 *) A[_, 1] = 1; A[n_, k_] := A[n, k] = If[k < 0, 0, n A[n, k - 1] + A[n, k - 2]];
T[n_, k_] := Sum[A[i+1, n] Binomial[k-1, i] (-1)^(i - k - 1), {i, 0, k-1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 01 2020, from Maple *)
PROG
(Sage)
@CachedFunction
flatten([[A117936(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 23 2021
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Apr 03 2006
STATUS
approved