Real-rooted atlas

A038320 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 17\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 408\,x + 289\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144408289841
11728734410404491324389
22073611750424969623582483521707281
32488321762560499392070747205011260141985720511149
42985984253808648989056016979328018040536010222970424137569594823321
5358318083553320961510161408356565888050513500804293647568202755579641033867317249876309
64299816964873125888241625825286846065049612123240192013739672217697322678208393925126086975757441500246412961
75159780352657871994883727941304321232291708928261861988147237097114987523503616415488212719568083275338180362811858787649714507145975869
86191736422487716265984055919119564802112500072448052372397629440890330759700481051084924646408508782723328045202908217680142305451796402015993900449420707233300201
9743008370688115785471098888201470869504034856251195392098759378386944019587276713410562774864201066496280789829869824019889279615779209392159818562402661111948592683427189630763312200509765705829
10891610044825615157370762035211810118052085765577000191262720177766881096499204029382638187315266596740825595904808674710025093127160140661680512045082367129099520191600060298672964935153068299152582622237229761353814783205469041
1110699320537907219704581990645761674889469204889687001202983698432308129260567265280785729614446526464148415593839899443221025542460652421122233963886444319744175821231803488128099632031355309939238494193932733385690889069007842716990457803290593710260628712958602189
12128391846454886425464382880219136234484525688684544132874564556921241651765715775300567041466695280300182732831167274706378883072504613019055658106886255098882044095283259075933885972011008418454531692301744642155674860233069593676346817966587881441663969109528197416168377826559400929297558232675799463481
13154070215745863683273992084599603203246708817227939840199311846835381862408470753490503729152026400515045403289190462334549412757766144011353792928752307404801608453998239910215680177227801657916033024015064363140922862807049700536871048813171204580809077995272886401497572198575377674403030800878069216722028624230515098157938629188747598184440949
1418488425889503641641907098682874920964452629235055460352029436826609533321216013553205584805966643204608089898834028658688119682334872494910996482422142491467158913024038602895957757845176320486110541690283976294404820596205095316098252837250061584827442577408219878835743773098547209584462070882417116160290956884294644805312054958522588988463225648661191875666868481250246473680347348787521
Technical details
Node id
n-bd9b770a0539d41d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8b732aeffa021ba5

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:          144,            408,            289;
  1:         1728,           7344,          10404,            4913;
  2:        20736,         117504,         249696,          235824,           83521;
  3:       248832,        1762560,        4993920,         7074720,         5011260,          1419857;
  4:      2985984,       25380864,       89890560,       169793280,       180405360,        102229704,         24137569;
  5:     35831808,      355332096,     1510161408,      3565658880,      5051350080,       4293647568,       2027555796,        410338673;
  6:    429981696,     4873125888,    24162582528,     68460650496,    121232401920,     137396722176,      97322678208,      39392512608,       6975757441;
  7:   5159780352,    65787199488,   372794130432,   1232291708928,   2618619881472,    3709711498752,    3503616415488,    2127195680832,     753381803628,    118587876497;
  8:  61917364224,   877162659840,  5591911956480,  21125000724480,  52372397629440,   89033075970048,  105108492464640,   85087827233280,   45202908217680,  14230545179640,   2015993900449;
  9: 743008370688, 11578547109888, 82014708695040, 348562511953920, 987593783869440, 1958727671341056, 2774864201066496, 2807898298698240, 1988927961577920, 939215981856240, 266111194859268, 34271896307633;
%F Formula
The row polynomials satisfy P(n,x) = (12+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 408*x + 289*x^2.
%t Mathematica
P[0, x_] := 144 + 408*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 17*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] = 144 + 408*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 17*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

Candidate equalities ?

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.