Real-rooted atlas

A207824 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-84\,n - 10\,n x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-210 - 88\,x - 10\,x^{2} - 210\,n - 109\,n x - 10\,n x^{2} + n x^{3} - 21\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3} + 5\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-441 - 315\,x - 71\,x^{2} - 5\,x^{3} + 21\,n x^{2} + 10\,n x^{3} + n x^{4} + 21\,n^{2} x^{2} + 10\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22420347
3115146454310
45519404418052017
5264056563570984125612981
612649326202589396001850180782799
760605182742174825815402112531142458524204
82903761002300112234563120020570040740485132093297841
9139127554095106940350457092017963754474267154071364051020634947
1066659992882780041693670314772001448982543520408748461171201004450011128509055
113193872015206148824476970020840664010999925038700564938007015782881815751365060512797030562
12153027601795357080141024706213369156007960147253211498809117616218494400267340526950018029720134925344177
1373319928541307791147999501230835650077655412188752522266506821175600195095040339916504304300385990232568451430338442481
1435129688242132493706044783427759511115899203735570562018939645300695734609518954672003867667655902050066466515366402940698015186334088735
Technical details
Node id
n-bcaca161d8d1b399 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1251e6617cbea0ee

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:       5,       2;
  2:      24,      20,       3;
  3:     115,     146,      45,       4;
  4:     551,     940,     441,      80,       5;
  5:    2640,    5656,    3570,     984,     125,      6;
  6:   12649,   32620,   25893,    9600,    1850,    180,     7;
  7:   60605,  182742,  174825,   81540,   21125,   3114,   245,    8;
  8:  290376, 1002300, 1122345,  631200,  205700,  40740,  4851,  320,   9;
  9: 1391275, 5409510, 6940350, 4570920, 1796375, 447426, 71540, 7136, 405, 10;
%F Formula
The row polynomials satisfy -(84*n+10*n*x-2*n^3*x^2)*P(n,x) = -(210*n+210+(21*n^2+109*n+88)*x-(5*n^3+5*n^2-10*n-10)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (441+315*x-(21*n^2+21*n-71)*x^2-(10*n^2+10*n-5)*x^3-(n^2+n)*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[(-(210*n + 210 + (21*n^2 + 109*n + 88)*x - (5*n^3 + 5*n^2 - 10*n - 10)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (441 + 315*x - (21*n^2 + 21*n - 71)*x^2 - (10*n^2 + 10*n - 5)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-84*n - 10*n*x + 2*n^3*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] = (-(210*n + 210 + (21*n^2 + 109*n + 88)*x - (5*n^3 + 5*n^2 - 10*n - 10)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (441 + 315*x - (21*n^2 + 21*n - 71)*x^2 - (10*n^2 + 10*n - 5)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-84*n - 10*n*x + 2*n^3*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023