A117936 Triangle, rows = inverse binomial transforms of A073133 columns.

1, 1, 1, 2, 3, 2, 3, 9, 12, 6, 5, 24, 56, 60, 24, 8, 62, 228, 414, 360, 120, 13, 156, 864, 2400, 3480, 2520, 720, 21, 387, 3132, 12606, 27360, 32640, 20160, 5040, 34, 951, 11034, 62220, 190704, 335160, 337680, 181440, 40320, 55, 2323, 38136, 294588, 1229760, 2997120, 4394880, 3820320, 1814400, 362880

Offset: 1

Author

Gary W. Adamson, Apr 03 2006

Keywords

• nonn
• tabl
• easy

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

Examples

			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;
  ...
		

Links

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

Crossrefs

Cf. A073133, A117937, A117938.

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

Programs

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 *)

Sage

@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

Formula

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