Real-rooted atlas

A113953 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
22439972
312841122527791
4704395138971080819713
538403629157789277834698243130644
620992325095788409586008167589191977291908019
711468828697551023290111052423475648290493275087218730008455

All 15 rows

Technical details
Node id
n-3322d2b2029a280d · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-661c52bda3ea3732

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:       4,         3;
  2:      24,        39,          9;
  3:     128,       411,        225,         27;
  4:     704,      3951,       3897,       1080,         81;
  5:    3840,     36291,      57789,      27783,       4698,        243;
  6:   20992,    325095,     788409,     586008,     167589,      19197,       729;
  7:  114688,   2869755,   10232901,   11052423,    4756482,     904932,     75087,     2187;
  8:  626688,  25109631,  128657025,  194346864,  118139877,   33198174,   4513968,   284310,    6561;
  9: 3424256, 218512179, 1583775693, 3261240711, 2691800586, 1048111245, 207635238, 21209526, 1049760, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (8+6*x)*P(n-2,x) + 6*x*d/dx P(n-2,x) + (4+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(8 + 6*x)*P[n - 2, x] + 6*x*D[P[n - 2, x], x] + (4 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 6*x)*v[i - 2] + 6*x*deriv(v[i - 2], x) + (4 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); 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