Real-rooted atlas

A008277 → θ+1 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
3108119
43749171104
5151284171341641
6674165814525286714380
7326399741159362921548132132803
8170076237590981665612513944432611266768

All 15 rows

Technical details
Node id
n-92978e16f285592e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4df902786f948216

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      3,       1;
   3:     10,       8,       1;
   4:     37,      49,      17,       1;
   5:    151,     284,     171,      34,       1;
   6:    674,    1658,    1452,     528,      67,       1;
   7:   3263,    9974,   11593,    6292,    1548,     132,      1;
   8:  17007,   62375,   90981,   66561,   25139,    4443,    261,     1;
   9:  94828,  406832,  717055,  665102,  344200,   96296,  12673,   518,    1;
  10: 562595, 2769493, 5740244, 6480624, 4297703, 1676293, 361050, 36202, 1031, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2+(n-3)*x)*P(n-2,x) + (1+2*x+x^2)*d/dx P(n-2,x) + (n+1+(n-1)*x)*P(n-1,x) - (1+2*x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-(n - 2 + (n - 3)*x)*P[n - 2, x] + (1 + 2*x + x^2)*D[P[n - 2, x], x] + (n + 1 + (n - 1)*x)*P[n - 1, x] - (1 + 2*x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(n - 2 + (n - 3)*x)*v[i - 2] + (1 + 2*x + x^2)*deriv(v[i - 2], x) + (n + 1 + (n - 1)*x)*v[i - 1] - (1 + 2*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.