OFFSET
0,4
COMMENTS
EXAMPLE
G.f. of row n = (1)(1+x)(2+x)(3+2x)(5+3x)*...*[F(n+1) + F(n)*x]:
row 3 g.f.: (1+x)(2+x)(3+2x) = 6 + 13x + 9x^2 + 2x^3;
row 4 g.f.: (1+x)(2+x)(3+2x)(5+3x) = 30 + 83x + 84x^2 + 37x^3 + 6x^4.
Triangle begins:
1;
1, 1;
2, 3, 1;
6, 13, 9, 2;
30, 83, 84, 37, 6;
240, 814, 1087, 716, 233, 30;
3120, 12502, 20643, 18004, 8757, 2254, 240;
65520, 303102, 596029, 646443, 417949, 161175, 34342, 3120;
...
Row vectors equal the product of flipped submatrices of Pascal's triangle;
for example, row vector 3 is equal to the matrix product:
[1 0 0 0] [1 1 0 0] [1 2 1 0] [1 3 3 1] = [6 13 9 2];
[0 0 0 0] [1 0 0 0] [1 1 0 0] [1 2 1 0]
[0 0 0 0] [0 0 0 0] [1 0 0 0] [1 1 0 0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [1 0 0 0]
likewise, row 4 may be obtained by the product:
[6 13 9 2 0] * [1 4 6 4 1] = [30 83 84 37 6] .
............. [1 3 3 1 0]
............. [1 2 1 0 0]
............. [1 1 0 0 0]
............. [1 0 0 0 0]
PROG
(PARI) {T(n, k)=polcoeff(prod(i=0, n, round((fibonacci(i+1)+x*fibonacci(i)))), k)}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, May 24 2007
STATUS
approved