Real-rooted atlas

A038221 → basis A013611 → basis A008288

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-108 + 72\,x\right)\,P_{n-2}(x) + \left(24 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 18 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1181230
23245761441044
358321771211232172836504
4104976456192508032186624207361276560
51889568107153281772928011197440286156824883244642016
6340122242381322245320650244979128322142443524180377629859841561156416
7612220032510211353614483888640184018728961161398476837533818885912248323583180854594518400
8110199605761065279651843685211873285999857090565122694430722403597680646191736422481696522244299816961909201031424
9198359290368218196227174489259362795521787734897459219564919635968123467094097924576725172224977778376704110935277568515978035266765834468864
1035704672266244403582292787220827574323200049790267965440067196013015859254202460641689626862074501529681963110891520149375641190401486016741376619173642242334838803747840
1164268410079232878335300546560472014787807641613160886953361408212760330851450882114734171344076813345869916274688541448774929612814004779027005442223452549283841968972182323274300837068881650626894043136
1211568313814261761735247289817497610450037166214348833372542552761958463183273636180787275269107002271334458408370975224627230058682264322048010287340971569971223057035759190016324100251294105625856691299942489161004482562855368371250630656
132082296486567116834010843919998976022697027654766428168181889528867061760178184630664611758082489453999396565811223131556526736539648145992262175120424966314595209353101312186467089627113062436835085781880012846435051134517248337028596944076810699320537907299853838796837150720
143748133675820810246621702905647595520485175055509964062721950834626812206120964816160830523863203847754918381083070300168450045529576576122886375507577978214154243372729226156352471041254439696602822082563252654964616881766457400784753818337286557613557683322884365322779466137612839184645488643491944935321114722304
Technical details
Node id
n-d5b6cf9ef2f79c87 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

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:           18,            12;
  2:          324,           576,           144;
  3:         5832,         17712,         11232,           1728;
  4:       104976,        456192,        508032,         186624,          20736;
  5:      1889568,      10715328,      17729280,       11197440,        2861568,         248832;
  6:     34012224,     238132224,     532065024,      497912832,      214244352,       41803776,       2985984;
  7:    612220032,    5102113536,   14483888640,    18401872896,    11613984768,     3753381888,     591224832,     35831808;
  8:  11019960576,  106527965184,  368521187328,   599985709056,   512269443072,   240359768064,   61917364224,   8169652224,    429981696;
  9: 198359290368, 2181962271744, 8925936279552, 17877348974592, 19564919635968, 12346709409792, 4576725172224, 977778376704, 110935277568, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = -(108-72*x)*P(n-2,x) + (24+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 18 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 18 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(108 - 72*x)*P[n - 2, x] + (24 + 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] = 18 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(108 - 72*x)*v[i - 2] + (24 + 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

  • 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.