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.

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.

Links

Table of n, a(n) for n=0..41.

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: A247500 A075263 A130850 * A058372 A128264 A232435

Adjacent sequences: A130402 A130403 A130404 * A130406 A130407 A130408

Keyword

nonn,tabl

Author

Paul D. Hanna, May 24 2007

Status

approved