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A130405
Triangle where g.f. of row n = Product_{i=0..n} [F(i+1) + F(i)*x] for n>=0, where F(i) = A000045(i) is the i-th Fibonacci number.
3
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, 2227680, 11681388, 26630128, 34495671, 27785569, 14256879
OFFSET
0,4
COMMENTS
First column equals A003266, the product of the first n nonzero Fibonacci numbers. Main diagonal and row sums are shifted forms of A003266.
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
Cf. A130406 (column 0), A130407 (diagonal); A003266, A000045.
Sequence in context: A375504 A075263 A130850 * A058372 A128264 A375258
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, May 24 2007
STATUS
approved