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

1,4

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]

Links

Table of n, a(n) for n=1..52.

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);...

Example

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.

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

Crossrefs

Cf. A117936, A117938.

Sequence in context: A050610 A151848 A238238 * A110897 A116644 A166462

Adjacent sequences: A117934 A117935 A117936 * A117938 A117939 A117940

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Apr 04 2006

Status

approved