Real-rooted atlas

A033282 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-12 + 162\,x - 54\,x^{2} + 48\,n + 81\,n x - 27\,n x^{2} + 81\,n^{2} + 27\,n^{3}\right)\,P_{n}(x) = \left(216 + 738\,x + 486\,x^{2} - 324\,x^{3} + 594\,n + 1341\,n x + 243\,n x^{2} - 162\,n x^{3} + 405\,n^{2} + 810\,n^{2} x + 81\,n^{3} + 162\,n^{3} x\right)\,P_{n-1}(x) + \left(28 + 162\,x + 252\,x^{2} - 162\,x^{4} + 72\,n + 324\,n x + 324\,n x^{2} + 36\,n^{2} + 162\,n^{2} x + 162\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
25613545236
3324141317013783816
41835117182545220412340262819
51032387180277020400140240570320761047309
657982610173256365054128255853870281466931274117625910
7325818410959521517650600519159486477081972891328378053127410298807854

All 15 rows

Technical details
Node id
n-342b3ea85f10e01a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7eee00ef1f81161d

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:        9,         6;
  2:       56,       135,         45;
  3:      324,      1413,       1701,        378;
  4:     1835,     11718,      25452,      20412,        3402;
  5:    10323,     87180,     277020,     400140,      240570,       32076;
  6:    57982,    610173,    2563650,    5412825,     5853870,     2814669,      312741;
  7:   325818,   4109595,   21517650,   60051915,    94864770,    81972891,    32837805,     3127410;
  8:  1833173,  26966412,  169266636,  588648060,  1228701474,  1549944396,  1115025912,   382794984,   31899582;
  9: 10330173, 173674962, 1271699676, 5301960300, 13788295794, 22974522480, 24116083992, 14859405288, 4463041518, 330595668;
%F Formula
The row polynomials satisfy (27*n^3+81*n^2+48*n-12+(81*n+162)*x-(27*n+54)*x^2)*P(n,x) = (81*n^3+405*n^2+594*n+216+(162*n^3+810*n^2+1341*n+738)*x+(243*n+486)*x^2-(162*n+324)*x^3)*P(n-1,x) + (36*n^2+72*n+28+(162*n^2+324*n+162)*x+(162*n^2+324*n+252)*x^2-162*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[((81*n^3 + 405*n^2 + 594*n + 216 + (162*n^3 + 810*n^2 + 1341*n + 738)*x + (243*n + 486)*x^2 - (162*n + 324)*x^3)*P[n - 1, x] + (36*n^2 + 72*n + 28 + (162*n^2 + 324*n + 162)*x + (162*n^2 + 324*n + 252)*x^2 - 162*x^4)*D[P[n - 1, x], x])/(27*n^3 + 81*n^2 + 48*n - 12 + (81*n + 162)*x - (27*n + 54)*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] = ((81*n^3 + 405*n^2 + 594*n + 216 + (162*n^3 + 810*n^2 + 1341*n + 738)*x + (243*n + 486)*x^2 - (162*n + 324)*x^3)*v[i - 1] + (36*n^2 + 72*n + 28 + (162*n^2 + 324*n + 162)*x + (162*n^2 + 324*n + 252)*x^2 - 162*x^4)*deriv(v[i - 1], x))/(27*n^3 + 81*n^2 + 48*n - 12 + (81*n + 162)*x - (27*n + 54)*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