Real-rooted atlas

A038322 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1272451
272912965762601
319683524884665613824132651
45314411889568251942414929923317766765201
51434890763772920113374080100776960447897607962624345025251
63874204892066242608459165024054419558403627970560128994508819110297617596287801
71046035320365086642152173564379072257132413440228562145280121899810816361184624644586471424897410677851
82824295364812008387814976624831764659211108120260608123423558451208776786378752390079394611299067782758411007531417645767944570401
97625597484987610047798798962169058840174084498788705546245998384940728325331897725091843159643096350721203673560514562674830134476826418075402242334165173090451
10205891132094649183014339639688073205735855875201735247072139264026992732233277440287922477154959362132759090036736010833062044631040361102068154368071328803586048063403380965376119042423827613001
115559060566555523543552588729873362415789283243881606442104755317017601145263067611914240142521626191704883212668588994818211848043548568138547203574910474728243201059232733252812801883080414671667215216811431690246071163615208263051
121500946352969991211601009443167990624782715727771017638423191577119141263360463831542382825267206596715269444626022468410380572018343936521221947215377858562895677484529876992011439713519130378240305059027176810086449302469038676377636520347436056576309629344375621415601
13405255515301897626746829526212663725752249757473134206537344814024356881858343936180894301529301854208028943088244688296673283430291940111205531648304914839120996047257620327655941399736483841003834861303690690560356919061796867801088865258331628770426881281864195005585817687648833846535782415791096563156692195651
14109418989131512359209136165853141437602571278673604037275059263362797283699103113218252868378045978076100890624121560970627690846027776162081294170254461370368164654013125337865519104128064232430818339848192758899155145590162063363372885133980400720281610902254978522507378688242272332856055719526433131259193990525747221035720123168587776805345924720991301978201
Technical details
Node id
n-f177c7ba598d3918 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-cb7716efe3e0f2cd

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:            27,             24;
  2:           729,           1296,             576;
  3:         19683,          52488,           46656,           13824;
  4:        531441,        1889568,         2519424,         1492992,          331776;
  5:      14348907,       63772920,       113374080,       100776960,        44789760,         7962624;
  6:     387420489,     2066242608,      4591650240,      5441955840,      3627970560,      1289945088,       191102976;
  7:   10460353203,    65086642152,    173564379072,    257132413440,    228562145280,    121899810816,     36118462464,      4586471424;
  8:  282429536481,  2008387814976,   6248317646592,  11108120260608,  12342355845120,   8776786378752,   3900793946112,    990677827584,   110075314176;
  9: 7625597484987, 61004779879896, 216905884017408, 449878870554624, 599838494072832, 533189772509184, 315964309635072, 120367356051456, 26748301344768, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (27+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(27 + 24*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; for(n = 1, N, my(i = n + 1); v[i] = (27 + 24*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

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.