0,4

First column equals A003266, the product of the first n nonzero Fibonacci numbers. Main diagonal and row sums are shifted forms of A003266.

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]

(PARI) {T(n, k)=polcoeff(prod(i=0, n, round((fibonacci(i+1)+x*fibonacci(i)))), k)}

Cf. A130406 (column 0), A130407 (diagonal); A003266, A000045.

Sequence in context: A135894 A130850 A075263 this_sequence A058372 A128264 A114858

Adjacent sequences: A130402 A130403 A130404 this_sequence A130406 A130407 A130408

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), May 24 2007