OFFSET
0,3
COMMENTS
The n-th row consists of the coefficients in the expansion of (-x)^n - (1 - x)*(((1 - x - sqrt(1 + 2*x - 3*x^2))/2)^n - ((1 - x + sqrt(1 + 2*x - 3*x^2))/2)^n)/sqrt(1 + 2*x - 3*x^2). - Franck Maminirina Ramaharo, Oct 13 2018
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Fibonacci Polynomial
FORMULA
From Franck Maminirina Ramaharo, Oct 13 2018: (Start)
G.f.: 1/((1 + x*y)*(1 - y + x*y - x*y^2 + x^2*y^2)).
E.g.f.: exp(-x*y) - (exp(y*(1 - x - sqrt(1 + 2*x - 3*x^2))/2) - exp(y*(1 - x + sqrt(1 + 2*x - 3*x^2))/2))*(1 - x)/sqrt(1 + 2*x - 3*x^2). (End)
EXAMPLE
Triangle begins:
1;
1, -2;
1, -2, 2;
1, -2, 1, -1;
1, -2, 0, 2, 0;
1, -2, -1, 5, -4, 0;
1, -2, -2, 8, -7, 2, 1;
1, -2, -3, 11, -9, 0, 3, -2;
1, -2, -4, 14, -10, -6, 12, -6, 2;
1, -2, -5, 17, -10, -16, 27, -15, 3, -1;
1, -2, -6, 20, -9, -30, 47, -24, 0, 4, 0;
1, -2, -7, 23, -7, -48, 71, -28, -18, 22, -8, 0;
....
MATHEMATICA
P[x_, n_]:= Sum[Binomial[n-k-1, k]*x^k*(1-x)^(n-k), {k, 0, n}];
Table[Coefficient[P[x, n], x, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Franck Maminirina Ramaharo, Oct 14 2018 *)
PROG
(Maxima) P(x, n) := sum(binomial(n - k - 1, k)*x^k*(1 - x)^(n - k), k, 0, n)$
create_list(ratcoef(expand(P(x, n)), x, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Oct 14 2018 */
(Sage)
def p(n, x): return sum( binomial(n-j-1, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 15 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 24 2006
EXTENSIONS
Edited by N. J. A. Sloane, May 26 2007
Edited by Franck Maminirina Ramaharo, Oct 14 2018
STATUS
approved