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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.

%I A130405 #3 Mar 30 2012 18:37:04
%S A130405 1,1,1,2,3,1,6,13,9,2,30,83,84,37,6,240,814,1087,716,233,30,3120,
%T A130405 12502,20643,18004,8757,2254,240,65520,303102,596029,646443,417949,
%U A130405 161175,34342,3120,2227680,11681388,26630128,34495671,27785569,14256879
%N 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.
%C A130405 First column equals A003266, the product of the first n nonzero Fibonacci numbers. Main diagonal and row sums are shifted forms of A003266.
%e A130405 G.f. of row n = (1)(1+x)(2+x)(3+2x)(5+3x)*...*[F(n+1) + F(n)*x]:
%e A130405 row 3 g.f.: (1+x)(2+x)(3+2x) = 6 + 13x + 9x^2 + 2x^3;
%e A130405 row 4 g.f.: (1+x)(2+x)(3+2x)(5+3x) = 30 + 83x + 84x^2 + 37x^3 + 6x^4.
%e A130405 Triangle begins:
%e A130405 1;
%e A130405 1, 1;
%e A130405 2, 3, 1;
%e A130405 6, 13, 9, 2;
%e A130405 30, 83, 84, 37, 6;
%e A130405 240, 814, 1087, 716, 233, 30;
%e A130405 3120, 12502, 20643, 18004, 8757, 2254, 240;
%e A130405 65520, 303102, 596029, 646443, 417949, 161175, 34342, 3120;
%e A130405 ...
%e A130405 Row vectors equal the product of flipped submatrices of Pascal's triangle;
%e A130405 for example, row vector 3 is equal to the matrix product:
%e A130405 [1 0 0 0] [1 1 0 0] [1 2 1 0] [1 3 3 1] = [6 13 9 2];
%e A130405 [0 0 0 0] [1 0 0 0] [1 1 0 0] [1 2 1 0]
%e A130405 [0 0 0 0] [0 0 0 0] [1 0 0 0] [1 1 0 0]
%e A130405 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1 0 0 0]
%e A130405 likewise, row 4 may be obtained by the product:
%e A130405 [6 13 9 2 0] * [1 4 6 4 1] = [30 83 84 37 6] .
%e A130405 ............. [1 3 3 1 0]
%e A130405 ............. [1 2 1 0 0]
%e A130405 ............. [1 1 0 0 0]
%e A130405 ............. [1 0 0 0 0]
%o A130405 (PARI) {T(n,k)=polcoeff(prod(i=0,n,round((fibonacci(i+1)+x*fibonacci(i)))),k)}
%Y A130405 Cf. A130406 (column 0), A130407 (diagonal); A003266, A000045.
%K A130405 nonn,tabl
%O A130405 0,4
%A A130405 _Paul D. Hanna_, May 24 2007