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A158472 Triangle read by rows: n-th row is the expansion of the polynomial (x-F1)*(x-F2)*(x-F3)*...*(x-Fn). 4

%I #16 Apr 22 2019 22:14:41

%S 1,1,-1,1,-2,1,1,-4,5,-2,1,-7,17,-17,6,1,-12,52,-102,91,-30,1,-20,148,

%T -518,907,-758,240,1,-33,408,-2442,7641,-12549,10094,-3120,1,-54,1101,

%U -11010,58923,-173010,273623,-215094,65520

%N Triangle read by rows: n-th row is the expansion of the polynomial (x-F1)*(x-F2)*(x-F3)*...*(x-Fn).

%C Row sums of the unsigned triangle = A082480: (1, 2, 4, 12, 48, 288, 2592, ...).

%C Right border starting with row 1 (unsigned) = A003266: (1, 1, 2, 6, 30, 240, ...).

%H Alois P. Heinz, <a href="/A158472/b158472.txt">Rows n = 0..98, flattened</a>

%e First few rows of the unsigned triangle:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 4, 5, 2;

%e 1, 7, 17, 17, 6;

%e 1, 12, 52, 102, 91, 30;

%e 1, 20, 148, 518, 907, 758, 240;

%e 1, 33, 408, 2442, 7641, 12549, 10094, 3120;

%e 1, 54, 1101, 11010, 58923, 173010, 273623, 215094, 65520;

%e ...

%e Example: row 5 is x^5 - 12x^4 + 52x^3 - 102x^2 + 91x - 30

%e = (x-1)*(x-1)*(x-2)*(x-3)*(x-5).

%p p:= proc(n) option remember; expand(`if`(n=0, 1,

%p p(n-1)*(x-(<<0|1>, <1|1>>^n)[1, 2])))

%p end:

%p T:= (n, k)-> coeff(p(n), x, n-k):

%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Nov 06 2016

%t Array[Reverse@ CoefficientList[Times @@ Array[(x - Fibonacci@ #) &, #], x] &, 9, 0] // Flatten (* _Michael De Vlieger_, Apr 21 2019 *)

%o (PARI) row(n) = Vec(prod(k=1, n, x-fibonacci(k)));

%o for (n=0, 10, print(row(n))); \\ _Michel Marcus_, Apr 22 2019

%Y Cf. A000045, A082480, A003266.

%K tabl,sign

%O 0,5

%A _Gary W. Adamson_, Mar 20 2009

%E One term corrected by _Alois P. Heinz_, Nov 06 2016

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)