Real-rooted atlas

A028246 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2347
37241849
41510018096391
531360117014406003601
663120463001344012600432037927
712738643061810080015960012096035280451249

All 15 rows

Technical details
Node id
n-35fb1e5d883b2099 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-19b75a1fb23b2bbe

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:   0;
  1:   1;
  2:   3,     4;
  3:   7,    24,     18;
  4:  15,   100,    180,      96;
  5:  31,   360,   1170,    1440,      600;
  6:  63,  1204,   6300,   13440,    12600,     4320;
  7: 127,  3864,  30618,  100800,   159600,   120960,    35280;
  8: 255, 12100, 139860,  667296,  1587600,  1995840,  1270080,   322560;
  9: 511, 37320, 613890, 4082400, 13696200, 25401600, 26460000, 14515200, 3265920;
%F Formula
The row polynomials satisfy P(n,x) = -(2+4*x+2*x^2)*P(n-2,x) - (x+2*x^2+x^3)*d/dx P(n-2,x) + (3+4*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(2 + 4*x + 2*x^2)*P[n - 2, x] - (x + 2*x^2 + x^3)*D[P[n - 2, x], x] + (3 + 4*x)*P[n - 1, x] + (x + x^2)*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 4*x + 2*x^2)*v[i - 2] - (x + 2*x^2 + x^3)*deriv(v[i - 2], x) + (3 + 4*x)*v[i - 1] + (x + x^2)*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

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.