Real-rooted atlas

A091044 → basis A007318 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-n - 3\,n x + 2\,n x^{2} + n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(x^{2} - 2\,x^{3} - 8\,n - 17\,n x + n x^{2} + 6\,n x^{3} + 12\,n^{2} + 21\,n^{2} x + 12\,n^{2} x^{2} + 8\,n^{3} + 6\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 6\,x^{3} - 4\,x^{4} - 8\,n x - 16\,n x^{2} - 8\,n x^{3} - 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
23222357
3160176524392
4768118457610052633
53584716849921456170617376
61638440448371201574431362667113105
7737282170882488321408004128060483928728176
8327680112230415482881104896436480950401075255294646001
9144179256360969109504788480039603201171456198528179527501029421208
106291456276561925131468852363264321126401216051228188163843842851299011185131465
1127262976133169152279183360328728576238977024111132672331735046223360700128434721276121158595512
1211744051263124275214763950081972633600166232064092081356833924710482374656128128001212640640641612137216558969
13503316480295279001676252446721140745830410950410240706019328031117148169366896641894164482489344020134409172820021444764234544
14214748364813656653824386043740166396313600068975329280508361441282617501286494621532162381414400408478720460528643224832128128245015276659588385
Technical details
Node id
n-9ca7cb1ef72efb17 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-dd5843b5797216f1

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       6,       2;
  2:      32,      22,       3;
  3:     160,     176,      52,       4;
  4:     768,    1184,     576,     100,       5;
  5:    3584,    7168,    4992,    1456,     170,       6;
  6:   16384,   40448,   37120,   15744,    3136,     266,      7;
  7:   73728,  217088,  248832,  140800,   41280,    6048,    392,     8;
  8:  327680, 1122304, 1548288, 1104896,  436480,   95040,  10752,   552,   9;
  9: 1441792, 5636096, 9109504, 7884800, 3960320, 1171456, 198528, 17952, 750, 10;
%F Formula
The row polynomials satisfy (2*n^3+n^2-n+(4*n^2-3*n)*x+2*n*x^2)*P(n,x) = (8*n^3+12*n^2-8*n+(6*n^3+21*n^2-17*n)*x+(12*n^2+n+1)*x^2+(6*n-2)*x^3)*P(n-1,x) - ((4*n^2+8*n)*x+(4*n^2+16*n+2)*x^2+(8*n+6)*x^3+4*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 + 12*n^2 - 8*n + (6*n^3 + 21*n^2 - 17*n)*x + (12*n^2 + n + 1)*x^2 + (6*n - 2)*x^3)*P[n - 1, x] - ((4*n^2 + 8*n)*x + (4*n^2 + 16*n + 2)*x^2 + (8*n + 6)*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n^3 + n^2 - n + (4*n^2 - 3*n)*x + 2*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^3 + 12*n^2 - 8*n + (6*n^3 + 21*n^2 - 17*n)*x + (12*n^2 + n + 1)*x^2 + (6*n - 2)*x^3)*v[i - 1] - ((4*n^2 + 8*n)*x + (4*n^2 + 16*n + 2)*x^2 + (8*n + 6)*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n^3 + n^2 - n + (4*n^2 - 3*n)*x + 2*n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023