Real-rooted atlas

A013610 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 2\,n - 3\,n x + 3\,n^{3} x\right)\,P_{n}(x) = \left(2 + 36\,x - 54\,x^{2} - 2\,n - 39\,n x - 27\,n x^{2} + 54\,n^{2} x^{2} + 3\,n^{3} x + 27\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x + 48\,x^{2} - 108\,x^{3} + 18\,n x^{2} - 162\,n x^{3} + 6\,n^{2} x^{2} - 54\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
211819
315455
41108243352
5118012151396
61270364529166832
7137885052041229296
81504170108164832805131968
9164830618244944295245571456
1018105103061236014762253542942494720
1119908019013471925412825389723410738432
121118812028526943841623847523383404372008746157824
131140417374550038564222003510132808448361131197088256
141163824324387567489851341535464829433852791738263752838955008
Technical details
Node id
n-98af4315ab9bc8aa · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-34767a487aafceeb

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 1;
  2: 1,  18;
  3: 1,  54;
  4: 1, 108,   243;
  5: 1, 180,  1215;
  6: 1, 270,  3645,   2916;
  7: 1, 378,  8505,  20412;
  8: 1, 504, 17010,  81648,  32805;
  9: 1, 648, 30618, 244944, 295245;
%F Formula
The row polynomials satisfy -(2*n-2-(3*n^3-3*n)*x)*P(n,x) = -(2*n-2-(3*n^3-39*n+36)*x-(27*n^3+54*n^2-27*n-54)*x^2)*P(n-1,x) - (4*x-(6*n^2+18*n+48)*x^2+(54*n^2+162*n+108)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 - (3*n^3 - 39*n + 36)*x - (27*n^3 + 54*n^2 - 27*n - 54)*x^2)*P[n - 1, x] - (4*x - (6*n^2 + 18*n + 48)*x^2 + (54*n^2 + 162*n + 108)*x^3)*D[P[n - 1, x], x])/(-2*n + 2 + (3*n^3 - 3*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 2 - (3*n^3 - 39*n + 36)*x - (27*n^3 + 54*n^2 - 27*n - 54)*x^2)*v[i - 1] - (4*x - (6*n^2 + 18*n + 48)*x^2 + (54*n^2 + 162*n + 108)*x^3)*deriv(v[i - 1], x))/(-2*n + 2 + (3*n^3 - 3*n)*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.