A117937 Triangle, rows = inverse binomial transforms of A117938 columns.

1, 1, 1, 3, 3, 2, 4, 10, 12, 6, 7, 27, 58, 60, 24, 11, 71, 240, 420, 360, 120, 18, 180, 920, 2460, 3504, 2520, 720, 29, 449, 3360, 13020, 27720, 32760, 20160, 5040, 47, 1107, 11898, 64620, 194184, 337680, 338400, 181440, 40320, 76, 2710, 41268, 307194, 1257120, 3029760, 4415040

Offset: 1

Author

Gary W. Adamson, Apr 04 2006

Keywords

• nonn
• tabl
• easy

Comments

A117936 is the companion triangle using analogous Fibonacci polynomials. Left border of A117936 = the Lucas numbers; right border = factorials.

[Note that most of the comments here and in many related sequences by the same author refer to some unusual definition of binomial transforms for sequences starting at index 1. - R. J. Mathar, Jul 05 2012]

Examples

			First few rows of the triangle are:
1;
1, 1;
3, 3, 2;
4, 10, 12, 6;
7, 27, 58, 60, 24;
11, 71, 240, 420, 360, 120;
...
For example, row 4: (4, 10, 12, 6) = the inverse binomial transform of column 4 of A117938: (4, 14, 36, 76, 140...), being f(x), x =1,2,3...using the Lucas polynomial x^3 + 3x.
		

Crossrefs

Cf. A117936, A117938.

Programs

Maple

A117937 := proc(n,k)
    add( A117938(n+i,n)*binomial(k-1,i)*(-1)^(1+i-k),i=0..k-1) ;
end proc:
seq(seq(A117937(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Aug 16 2019

Formula

Rows of the triangle are inverse binomial transforms of A117938 columns. A117938 columns are generated from f(x), Lucas polynomials: (1); (x); (x^2 + 2); (x^3 + 3x); (x^4 + 4x + 2);...