The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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