This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A123018 #26 Feb 16 2025 08:33:02
%S A123018 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,
%T A123018 -2,-3,11,-9,0,3,-2,1,-2,-4,14,-10,-6,12,-6,2,1,-2,-5,17,-10,-16,27,
%U A123018 -15,3,-1,1,-2,-6,20,-9,-30,47,-24,0,4,0,1,-2,-7,23,-7
%N 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).
%C A123018 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
%H A123018 G. C. Greubel, <a href="/A123018/b123018.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A123018 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci Polynomial</a>
%F A123018 From _Franck Maminirina Ramaharo_, Oct 13 2018: (Start)
%F A123018 G.f.: 1/((1 + x*y)*(1 - y + x*y - x*y^2 + x^2*y^2)).
%F A123018 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)
%e A123018 Triangle begins:
%e A123018 1;
%e A123018 1, -2;
%e A123018 1, -2, 2;
%e A123018 1, -2, 1, -1;
%e A123018 1, -2, 0, 2, 0;
%e A123018 1, -2, -1, 5, -4, 0;
%e A123018 1, -2, -2, 8, -7, 2, 1;
%e A123018 1, -2, -3, 11, -9, 0, 3, -2;
%e A123018 1, -2, -4, 14, -10, -6, 12, -6, 2;
%e A123018 1, -2, -5, 17, -10, -16, 27, -15, 3, -1;
%e A123018 1, -2, -6, 20, -9, -30, 47, -24, 0, 4, 0;
%e A123018 1, -2, -7, 23, -7, -48, 71, -28, -18, 22, -8, 0;
%e A123018 ....
%t A123018 P[x_, n_]:= Sum[Binomial[n-k-1, k]*x^k*(1-x)^(n-k), {k, 0, n}];
%t A123018 Table[Coefficient[P[x, n], x, k], {n,0,12}, {k,0,n}]//Flatten (* _Franck Maminirina Ramaharo_, Oct 14 2018 *)
%o A123018 (Maxima) P(x, n) := sum(binomial(n - k - 1, k)*x^k*(1 - x)^(n - k), k, 0, n)$
%o A123018 create_list(ratcoef(expand(P(x, n)), x, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Oct 14 2018 */
%o A123018 (Sage)
%o A123018 def p(n,x): return sum( binomial(n-j-1, j)*x^j*(1-x)^(n-j) for j in (0..n) )
%o A123018 def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A123018 flatten([T(n) for n in (0..12)]) # _G. C. Greubel_, Jul 15 2021
%Y A123018 Row sums: A033999.
%Y A123018 Cf. A049310, A168561, A320508.
%Y A123018 Cf. A122753, A123019, A123021, A123027, A123199, A123202, A123217, A123221.
%K A123018 sign,tabl,easy
%O A123018 0,3
%A A123018 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 24 2006
%E A123018 Edited by _N. J. A. Sloane_, May 26 2007
%E A123018 Edited by _Franck Maminirina Ramaharo_, Oct 14 2018