Real-rooted atlas

A038326 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 12\,n x\right)\,P_{n}(x) = \left(121 + 264\,x + 144\,x^{2} + 132\,n x + 144\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112435
21215284321081
31331871214256691231211
414641127776313632304128103680863857
51610511756920574992083635205702400149299223226803
6177156123191344948736801839974401881792009853747220901888611452585
7194871712976222481461054672354195072048299328003793692672160944537628665446415839840123
82143588813741536832214288018566233833267210625852160011128165171270815596544252255928323869835264405174228193
9235794769146301518296303064483392102858248908821039187276802754220879872233691468595212486668451843831136911365159780352010258739071811
102593742460156590744584041671366466401616343911424038571843340800605928593571846426515386368045784450990080210712530124805675758387200681091006464257564830589209
112853116706116847480094664560248371381602444720166028806667418634624001221955997036544155521672350105613849796424499208498738715033603433833824256008241201178214489161004482566421109238115595
12313842837672182169761135968739527850223712358558957684224011001240747129600230425988012605443421476791702323236563462560677888280458377596108801510886882672640054391927776215041176925259169792115909305827328159119229571625809
1334522712143931979189653536952961386205290825651273930948844032174797491871059200411886453572532224698958830304903168871429191029489664802110959924871168540142060555468800259268189066625024841501560306401281657503073330790414979048753070083922717773331637459
143797498335832411159963128036081612337789634565595271783503328381644826918813748143116807047834872241106944134549574833693859841917144220264877260820587514638071693312166363754651084390409981825279065063424431970800957285990412762773664647086082306773507972792321925877696823296096270945129865270153
Technical details
Node id
n-2ff8964283bcb6ae · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7a38cc12383be86e

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:         11,          24;
  2:        121,         528,          432;
  3:       1331,        8712,        14256,          6912;
  4:      14641,      127776,       313632,        304128,        103680;
  5:     161051,     1756920,      5749920,       8363520,       5702400,       1492992;
  6:    1771561,    23191344,     94873680,     183997440,     188179200,      98537472,      20901888;
  7:   19487171,   297622248,   1461054672,    3541950720,    4829932800,    3793692672,    1609445376,     286654464;
  8:  214358881,  3741536832,  21428801856,   62338332672,  106258521600,  111281651712,   70815596544,   25225592832,   3869835264;
  9: 2357947691, 46301518296, 303064483392, 1028582489088, 2103918727680, 2754220879872, 2336914685952, 1248666845184, 383113691136, 51597803520;
%F Formula
The row polynomials satisfy (11+12*n*x)*P(n,x) = (121+(132*n+264)*x+(144*n+144)*x^2)*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[Together[((121 + (132*n + 264)*x + (144*n + 144)*x^2)*P[n - 1, x])/(11 + 12*n*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] = ((121 + (132*n + 264)*x + (144*n + 144)*x^2)*v[i - 1])/(11 + 12*n*x)); 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.