Real-rooted atlas

A038281 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 24\,x + 12\,n x\right)\,P_{n}(x) = \left(121 + 528\,x + 432\,x^{2} + 132\,n x + 144\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 528\,x + 432\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01215284321081
11331871214256691231211
214641127776313632304128103680863857
31610511756920574992083635205702400149299223226803
4177156123191344948736801839974401881792009853747220901888611452585
5194871712976222481461054672354195072048299328003793692672160944537628665446415839840123
62143588813741536832214288018566233833267210625852160011128165171270815596544252255928323869835264405174228193
7235794769146301518296303064483392102858248908821039187276802754220879872233691468595212486668451843831136911365159780352010258739071811
82593742460156590744584041671366466401616343911424038571843340800605928593571846426515386368045784450990080210712530124805675758387200681091006464257564830589209
92853116706116847480094664560248371381602444720166028806667418634624001221955997036544155521672350105613849796424499208498738715033603433833824256008241201178214489161004482566421109238115595
10313842837672182169761135968739527850223712358558957684224011001240747129600230425988012605443421476791702323236563462560677888280458377596108801510886882672640054391927776215041176925259169792115909305827328159119229571625809
1134522712143931979189653536952961386205290825651273930948844032174797491871059200411886453572532224698958830304903168871429191029489664802110959924871168540142060555468800259268189066625024841501560306401281657503073330790414979048753070083922717773331637459
123797498335832411159963128036081612337789634565595271783503328381644826918813748143116807047834872241106944134549574833693859841917144220264877260820587514638071693312166363754651084390409981825279065063424431970800957285990412762773664647086082306773507972792321925877696823296096270945129865270153
1341772481694156511367099400899667601565950222848710160987023170765247616040378220622214675200116289275391978264576246674220528438743040395410995429630935040485277130754547056640457500325290482073600329400234209147092992178187955394880471040701952551555589734401903088144077553664031776981997584384002465123451933818882353345773881366211227
14459497298635721611604063297055609984196862313729552134401336277523497566003205922139024591485696001860628406271652233216434146628130052187750477324816884016716185601067609687660003524608011502865321589263564800966240687013498139443262722160298997925806083088591226844594831360111647837785883148288027963744157874257920043386172754035212288314303240121561907257326575624844117805505
Technical details
Node id
n-d5695c73bb22f8cc · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7a38cc12383be86e

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          121,           528,            432;
  1:         1331,          8712,          14256,            6912;
  2:        14641,        127776,         313632,          304128,          103680;
  3:       161051,       1756920,        5749920,         8363520,         5702400,          1492992;
  4:      1771561,      23191344,       94873680,       183997440,       188179200,         98537472,         20901888;
  5:     19487171,     297622248,     1461054672,      3541950720,      4829932800,       3793692672,       1609445376,        286654464;
  6:    214358881,    3741536832,    21428801856,     62338332672,    106258521600,     111281651712,      70815596544,      25225592832,      3869835264;
  7:   2357947691,   46301518296,   303064483392,   1028582489088,   2103918727680,    2754220879872,    2336914685952,    1248666845184,    383113691136,     51597803520;
  8:  25937424601,  565907445840,  4167136646640,  16163439114240,  38571843340800,   60592859357184,   64265153863680,   45784450990080,  21071253012480,   5675758387200,   681091006464;
  9: 285311670611, 6847480094664, 56024837138160, 244472016602880, 666741863462400, 1221955997036544, 1555216723501056, 1384979642449920, 849873871503360, 343383382425600, 82412011782144, 8916100448256;
%F Formula
The row polynomials satisfy (11+(12*n+24)*x)*P(n,x) = (121+(132*n+528)*x+(144*n+432)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 528*x + 432*x^2.
%t Mathematica
P[0, x_] := 121 + 528*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((121 + (132*n + 528)*x + (144*n + 432)*x^2)*P[n - 1, x])/(11 + (12*n + 24)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 528*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (132*n + 528)*x + (144*n + 432)*x^2)*v[i - 1])/(11 + (12*n + 24)*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.