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

0,3

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

Row sums: A033999.

Cf. A049310, A168561, A320508.

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

Sequence in context: A309474 A022828 A129406 * A336532 A100429 A049710

Adjacent sequences: A123015 A123016 A123017 * A123019 A123020 A123021

Keyword

sign,tabl,easy

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