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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). 14
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 (list; table; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A309474 A022828 A129406 * A336532 A100429 A049710
KEYWORD
sign,tabl,easy
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, May 26 2007
Edited by Franck Maminirina Ramaharo, Oct 14 2018
STATUS
approved

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Last modified June 24 15:21 EDT 2024. Contains 373679 sequences. (Running on oeis4.)