Real-rooted atlas

A099040 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2410620
3832401696
4168817214444464
5322246007764881202240
664544185632723160159232810816
712812805312119041561612032505689652224
825629441440039232652966816043680157442448252160
95126656375041204482434883219522789761530244828866881217536
1010241484894720350592835136133993614687681087680521472146336182725878784
112048327682334729789442687360507801667482886314752408230417382404391684992028385280
1240967168056422426439688223232179022082800275231743488259070081485939256904321307392136384137056256
13819215564813414406948864241592325958348810734067214314508814142707210232691252740736183504643865728372608661766144
141638433587231457281785241668632576189219840386099200592781440688636800602808960391668096183282048584280321136422410179843195289600
Technical details
Node id
n-461fdc822e0edcc1 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-75d07e1be1e68206

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:   2,    2;
  2:   4,   10,     6;
  3:   8,   32,    40,     16;
  4:  16,   88,   172,    144,     44;
  5:  32,  224,   600,    776,    488,    120;
  6:  64,  544,  1856,   3272,   3160,   1592,    328;
  7: 128, 1280,  5312,  11904,  15616,  12032,   5056,    896;
  8: 256, 2944, 14400,  39232,  65296,  68160,  43680,  15744,  2448;
  9: 512, 6656, 37504, 120448, 243488, 321952, 278976, 153024, 48288, 6688;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+(6*n-4)*x+(6*n-6)*x^2)*P(n-1,x) - (2*x+8*x^2+6*x^3)*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[((2*n + (6*n - 4)*x + (6*n - 6)*x^2)*P[n - 1, x] - (2*x + 8*x^2 + 6*x^3)*D[P[n - 1, x], x])/(n)]];
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] = ((2*n + (6*n - 4)*x + (6*n - 6)*x^2)*v[i - 1] - (2*x + 8*x^2 + 6*x^3)*deriv(v[i - 1], x))/(n)); 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

Same fingerprint ?

References

  1. [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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741