Real-rooted atlas

A146987 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(36 + 45\,x + 9\,x^{2} - 12\,n - 15\,n x - 3\,n x^{2}\right)\,P_{n-2}(x) + \left(45\,x + 18\,x^{2} - 15\,n x - 6\,n x^{2}\right)\,P_{n-2}'(x) + \left(9\,x^{2} - 3\,n x^{2}\right)\,P_{n-2}''(x) + \left(-24 - 12\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-12\,x + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 7 + 8\,x + x^{2}\)
\(P_{3}(x) = 26 + 55\,x + 18\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
278116
32655181100
4124443271411880
568039374011117810119908
6395236143572412788348622701130352
7234563330257831835781661664051938876411904388

All 15 rows

Technical details
Node id
n-919aa614b1337522 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-e10f38a72634eca7

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,        1;
  2:      7,        8,         1;
  3:     26,       55,        18,         1;
  4:    124,      443,       271,        41,         1;
  5:    680,     3937,      4011,      1178,       101,        1;
  6:   3952,    36143,     57241,     27883,      4862,      270,       1;
  7:  23456,   333025,    783183,    578166,    166405,    19388,     764,      1;
  8: 140224,  3054983,  10344601,  10950891,   4689426,   895527,   75072,   2231,    1;
  9: 840320, 27877177, 133076871, 194888758, 116949021, 32758131, 4466049, 283233, 6615, 1;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(12*n-36+(15*n-45)*x+(3*n-9)*x^2)*P(n-2,x) - ((15*n-45)*x+(6*n-18)*x^2)*d/dx P(n-2,x) - (3*n-9)*x^2*d^2/dx^2 P(n-2,x) + (8*n-24+(4*n-12)*x)*P(n-1,x) + (4*n-12)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 7 + 8*x + x^2, P(3,x) = 26 + 55*x + 18*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 7 + 8*x + x^2;
P[3, x_] := 26 + 55*x + 18*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 36 + (15*n - 45)*x + (3*n - 9)*x^2)*P[n - 2, x] - ((15*n - 45)*x + (6*n - 18)*x^2)*D[P[n - 2, x], x] - (3*n - 9)*x^2*D[P[n - 2, x], {x, 2}] + (8*n - 24 + (4*n - 12)*x)*P[n - 1, x] + (4*n - 12)*x*D[P[n - 1, x], x])/(n - 3)]];
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] = 2 + x; v[3] = 7 + 8*x + x^2; v[4] = 26 + 55*x + 18*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(12*n - 36 + (15*n - 45)*x + (3*n - 9)*x^2)*v[i - 2] - ((15*n - 45)*x + (6*n - 18)*x^2)*deriv(v[i - 2], x) - (3*n - 9)*x^2*deriv(deriv(v[i - 2], x), x) + (8*n - 24 + (4*n - 12)*x)*v[i - 1] + (4*n - 12)*x*deriv(v[i - 1], x))/(n - 3)); 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