Real-rooted atlas

A038221 → basis A013610 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-90 + 27\,x\right)\,P_{n-2}(x) + \left(21 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 15 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
115924
222535181657
333758991510372918198
450625193671190269634236561504549
57593753804651552217531164757282715904913989996
61139062570668531138312441114574743444638977971615531441387913293
7170859375126463931131463078673519497547198351495958192789584499119478296910756029042

All 15 rows

Technical details
Node id
n-8e5ab335b48f40f5 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-0fee5d4c7376c95e

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          15,            9;
  2:         225,          351,            81;
  3:        3375,         8991,          5103,           729;
  4:       50625,       193671,        190269,         63423,          6561;
  5:      759375,      3804651,       5522175,       3116475,        728271,         59049;
  6:    11390625,     70668531,     138312441,     114574743,      44463897,       7971615,       531441;
  7:   170859375,   1264639311,    3146307867,    3519497547,    1983514959,     581927895,     84499119,     4782969;
  8:  2562890625,  22042538991,   66914149653,   95648928327,   72421059393,   30555200295,   7179236469,   875283327,   43046721;
  9: 38443359375, 376754999571, 1354557546999, 2379050374923, 2298192689655, 1294630134075, 433867900959, 84845087091, 8910671247, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = -(90-27*x)*P(n-2,x) + (21+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 15 + 9*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 15 + 9*x;
P[n_, x_] := P[n, x] = Expand[-(90 - 27*x)*P[n - 2, x] + (21 + 9*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] = 15 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = -(90 - 27*x)*v[i - 2] + (21 + 9*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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.