Real-rooted atlas

A128908 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - n\right)\,P_{n-2}(x) + \left(-6 - 2\,x + 3\,n + n x\right)\,P_{n-1}(x) + \left(-2\,x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
\(P_{2}(x) = 3 + 5\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
23519
382210141
4219171161200
5553634361662311044
61441416247214163263115806
737754451334010802367557340134253
8987207416967376736362018228932501213549
92584785063557665192873264681014521657614316111402132
10676529586717876633394752276836211298562497043083621017319665980
111771111117278878416216368882244681811705758336061655663353886297686169771516
124636841687524371614413530499217599625211496302641701598889611011472788952540931001526034239
131213931560964921387105683390920813448319381083994702486916792130106065214622822218602142648549211514133189943
143178115839123710412487855082021312100717365229901826528542454423217790813323650438894799598140689332194367216131133776503346
Technical details
Node id
n-ef9f9881a6863a64 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-1c220806245eaa60

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:    1,     1;
  2:    3,     5,      1;
  3:    8,    22,     10,      1;
  4:   21,    91,     71,     16,      1;
  5:   55,   363,    436,    166,     23,      1;
  6:  144,  1416,   2472,   1416,    326,     31,     1;
  7:  377,  5445,  13340,  10802,   3675,    573,    40,    1;
  8:  987, 20741,  69673,  76736,  36201,   8228,   932,   50,  1;
  9: 2584, 78506, 355766, 519287, 326468, 101452, 16576, 1431, 61, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(n-2)*P(n-2,x) + (3*n-6+(n-2)*x)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 3 + 5*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 3 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2)*P[n - 2, x] + (3*n - 6 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 3 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2)*v[i - 2] + (3*n - 6 + (n - 2)*x)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

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