Real-rooted atlas

A099093 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8\,n x^{2} + 3\,n x^{3} - n^{2} - 2\,n^{2} x + 3\,n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-12\,x - 28\,x^{2} - 24\,x^{3} + 18\,x^{4} - 6\,n - 16\,n x - 8\,n x^{2} + 12\,n x^{3} + 8\,n^{2} + 16\,n^{2} x - 6\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(-8\,x^{2} - 14\,x^{3} + 6\,x^{4} + 9\,x^{5} - 4\,n x - 11\,n x^{2} + 6\,n x^{3} + 9\,n x^{4} + 4\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16612
26602793
30108378108594
405499919444053402
50097268048910145818144
60032498283888037908510392043
700064807452019828815309017496449874

All 15 rows

Technical details
Node id
n-53053890cf3b9d99 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-43f3ff4195d49336

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: 6,   6;
  2: 6,  60,  27;
  3: 0, 108, 378,  108;
  4: 0,  54, 999, 1944,   405;
  5: 0,   0, 972, 6804,  8910,   1458;
  6: 0,   0, 324, 9828, 38880,  37908,    5103;
  7: 0,   0,   0, 6480, 74520, 198288,  153090,   17496;
  8: 0,   0,   0, 1620, 73305, 473850,  933849,  594864,   59049;
  9: 0,   0,   0,    0, 36450, 619650, 2673972, 4146552, 2243862, 196830;
%F Formula
The row polynomials satisfy (n^3-n^2-2*n^2*x+(3*n^2-8*n)*x^2+3*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(16*n^2-16*n-12)*x+(6*n^2+8*n+28)*x^2-(12*n-24)*x^3-18*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(3*n^2-11*n-8)*x^2+(6*n-14)*x^3+(9*n+6)*x^4+9*x^5)*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^3 - 8*n^2 + 6*n - (16*n^2 - 16*n - 12)*x + (6*n^2 + 8*n + 28)*x^2 - (12*n - 24)*x^3 - 18*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (3*n^2 - 11*n - 8)*x^2 + (6*n - 14)*x^3 + (9*n + 6)*x^4 + 9*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 - 2*n^2*x + (3*n^2 - 8*n)*x^2 + 3*n*x^3)]];
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^3 - 8*n^2 + 6*n - (16*n^2 - 16*n - 12)*x + (6*n^2 + 8*n + 28)*x^2 - (12*n - 24)*x^3 - 18*x^4)*v[i - 1] + ((4*n^2 - 4*n)*x + (3*n^2 - 11*n - 8)*x^2 + (6*n - 14)*x^3 + (9*n + 6)*x^4 + 9*x^5)*deriv(v[i - 1], x))/(n^3 - n^2 - 2*n^2*x + (3*n^2 - 8*n)*x^2 + 3*n*x^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

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. [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
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.