A123018 Triangle read by rows: row n gives the coefficients of x^k (0 <= k <= n) in the expansion of Sum_{j=0..n} A320508(n,j)*x^j*(1 - x)^(n - j).

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

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

Keywords

• sign
• tabl
• easy

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

Examples

			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;
     ....
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Crossrefs

Row sums: A033999.

Cf. A049310, A168561, A320508.

Cf. A122753, A123019, A123021, A123027, A123199, A123202, A123217, A123221.

Programs

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 *)

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

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)

Extensions

Edited by N. J. A. Sloane, May 26 2007

Edited by Franck Maminirina Ramaharo, Oct 14 2018