Real-rooted atlas

A159854 → basis A008277 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 4\,x + x^{2} - x^{3} - n x - 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 5\,x + 5\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
015511
1216332071
244815820087497
38140664133012064093757
41640426167418105647439206630523
532116499643768675174826394747311155265287
664336437304181446476596728425650096314901640772456273
712897721386128463782817198562153369365095199506217468738994624134269

All 15 rows

Technical details
Node id
n-9320a39650137975 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5a6e7fab76a29249

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,     5,       5;
  1:   2,    16,      33,       20;
  2:   4,    48,     158,      200,       87;
  3:   8,   140,     664,     1330,     1206,       409;
  4:  16,   404,    2616,     7418,    10564,      7439,      2066;
  5:  32,  1164,    9964,    37686,    75174,     82639,     47473,     11155;
  6:  64,  3364,   37304,   181446,   476596,    728425,    650096,    314901,     64077;
  7: 128,  9772,  138612,   846378,  2817198,   5621533,   6936509,   5199506,   2174687,    389946;
  8: 256, 28548,  514208,  3876490, 15932904,  39906859,  63892214,  65928476,  42533391,  15637264,   2504665;
  9: 512, 83852, 1911532, 17582894, 87607270, 268145487, 535260865, 713283059, 631264386, 357024995, 116999102, 16923381;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2-(n-4)*x-(2*n-1)*x^2-(n+1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 5*x + 5*x^2.
%t Mathematica
P[0, x_] := 1 + 5*x + 5*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 - (n - 4)*x - (2*n - 1)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 5*x + 5*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 - (n - 4)*x - (2*n - 1)*x^2 - (n + 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(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. [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.