Real-rooted atlas

image under θ+1 → θ+1 → D+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 26\,x + 29\,x^{2} + 12\,x^{3} + x^{4} - 4\,n - 12\,n x - 14\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} + 2\,n^{2} x + 5\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 13\,x^{2} + 15\,x^{3} + 7\,x^{4} + x^{5} - n x^{2} - 3\,n x^{3} - 3\,n x^{4} - n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
20182745
3004864112
4000100125225
50000180216396
600000294343637
7000000448512960
800000006487291377
90000000090010001900
10000000000121013312541
110000000000158417283312
1200000000000202821974225
13000000000000254827445292
140000000000000315033756525
Technical details
Node id
n-8a3afca20b1dac35 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6367169308476412

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 4,  8;
  2: 0, 18, 27;
  3: 0,  0, 48,  64;
  4: 0,  0,  0, 100, 125;
  5: 0,  0,  0,   0, 180, 216;
  6: 0,  0,  0,   0,   0, 294, 343;
  7: 0,  0,  0,   0,   0,   0, 448, 512;
  8: 0,  0,  0,   0,   0,   0,   0, 648, 729;
  9: 0,  0,  0,   0,   0,   0,   0,   0, 900, 1000;
%F Formula
The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = -(4*n-8-(2*n^2-12*n+26)*x-(5*n^2-14*n+29)*x^2-(4*n^2-8*n+12)*x^3-(n^2-2*n+1)*x^4)*P(n-1,x) + (4*x-(n-13)*x^2-(3*n-15)*x^3-(3*n-7)*x^4-(n-1)*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[(-(4*n - 8 - (2*n^2 - 12*n + 26)*x - (5*n^2 - 14*n + 29)*x^2 - (4*n^2 - 8*n + 12)*x^3 - (n^2 - 2*n + 1)*x^4)*P[n - 1, x] + (4*x - (n - 13)*x^2 - (3*n - 15)*x^3 - (3*n - 7)*x^4 - (n - 1)*x^5)*D[P[n - 1, x], x])/(n^2 + 2*n^2*x + n^2*x^2)]];
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] = (-(4*n - 8 - (2*n^2 - 12*n + 26)*x - (5*n^2 - 14*n + 29)*x^2 - (4*n^2 - 8*n + 12)*x^3 - (n^2 - 2*n + 1)*x^4)*v[i - 1] + (4*x - (n - 13)*x^2 - (3*n - 15)*x^3 - (3*n - 7)*x^4 - (n - 1)*x^5)*deriv(v[i - 1], x))/(n^2 + 2*n^2*x + n^2*x^2)); 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

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [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
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.