Real-rooted atlas

A124932 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 27\,x + 9\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-27 - 18\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(9 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 3\,x\)
\(P_{2}(x) = 45 + 33\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18311
24533684
32162348410544
49451350720170153200
538886885486017103002117664
61530932319283501323034654832893184
7583201428841496888694030240630072836475136

All 15 rows

Technical details
Node id
n-ff2ca64442a7da44 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a1fb2ed15211d00c

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:      8,       3;
  2:     45,      33,       6;
  3:    216,     234,      84,      10;
  4:    945,    1350,     720,     170,      15;
  5:   3888,    6885,    4860,    1710,     300,     21;
  6:  15309,   32319,   28350,   13230,    3465,    483,     28;
  7:  58320,  142884,  149688,   86940,   30240,   6300,    728,    36;
  8: 216513,  603612,  734832,  510300,  221130,  61236,  10584,  1044,   45;
  9: 787320, 2460375, 3411720, 2755620, 1428840, 493290, 113400, 16740, 1440, 55;
%F Formula
The row polynomials satisfy P(n,x) = (27+27*x+9*x^2+x^3)*P(n-3,x) - (27+18*x+3*x^2)*P(n-2,x) + (9+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 8 + 3*x, P(2,x) = 45 + 33*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 3*x;
P[2, x_] := 45 + 33*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 27*x + 9*x^2 + x^3)*P[n - 3, x] - (27 + 18*x + 3*x^2)*P[n - 2, x] + (9 + 3*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] = 8 + 3*x; v[3] = 45 + 33*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (27 + 27*x + 9*x^2 + x^3)*v[i - 3] - (27 + 18*x + 3*x^2)*v[i - 2] + (9 + 3*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.