Real-rooted atlas

A124932 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 48\,x + 96\,x^{2} + 64\,x^{3}\right)\,P_{n-3}(x) + \left(-12 - 48\,x - 48\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 12\,x\)
\(P_{2}(x) = 18 + 84\,x + 96\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151217
2188496198
3563848646401944
4160144048007040384017280
5432480021120460804992021504143856
6112014784806402329603763203225601146881143072
728164300827955210035202150400275251219496965898248771328

All 15 rows

Technical details
Node id
n-2e4db12513fbf644 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-4a5b92b7fb302040

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:     5,     12;
  2:    18,     84,      96;
  3:    56,    384,     864,      640;
  4:   160,   1440,    4800,     7040,     3840;
  5:   432,   4800,   21120,    46080,    49920,    21504;
  6:  1120,  14784,   80640,   232960,   376320,   322560,    114688;
  7:  2816,  43008,  279552,  1003520,  2150400,  2752512,   1949696,    589824;
  8:  6912, 119808,  903168,  3870720, 10321920, 17547264,  18579456,  11206656,  2949120;
  9: 16640, 322560, 2764800, 13762560, 43868160, 92897280, 130744320, 117964800, 61931520, 14417920;
%F Formula
The row polynomials satisfy P(n,x) = (8+48*x+96*x^2+64*x^3)*P(n-3,x) - (12+48*x+48*x^2)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 12*x, P(2,x) = 18 + 84*x + 96*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 12*x;
P[2, x_] := 18 + 84*x + 96*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 48*x + 96*x^2 + 64*x^3)*P[n - 3, x] - (12 + 48*x + 48*x^2)*P[n - 2, x] + (6 + 12*x)*P[n - 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; v[2] = 5 + 12*x; v[3] = 18 + 84*x + 96*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 48*x + 96*x^2 + 64*x^3)*v[i - 3] - (12 + 48*x + 48*x^2)*v[i - 2] + (6 + 12*x)*v[i - 1]); 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.