1,4

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

G. C. Greubel, Rows n = 1..50 of the triangle, flatten

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 + 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;

...

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

(* 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 *)

(SageMath)

@CachedFunction

def A073133(n, k): return 0 if (k<0) else 1 if (k==1) else n*A073133(n, k-1) + A073133(n, k-2)

def A117936(n, k): return sum( (-1)^(j-k+1)*binomial(k-1, j)*A073133(j+1, n) for j in (0..k-1) )

flatten([[A117936(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 23 2021

Cf. A073133, A117937, A117938.

Cf. A006684 (column 2), A309717 (column 3 halved).

Sequence in context: A300663 A102310 A151546 * A264766 A251090 A078331

Adjacent sequences: A117933 A117934 A117935 * A117937 A117938 A117939

nonn,tabl,easy

Gary W. Adamson, Apr 03 2006

approved