Real-rooted atlas

A038335 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
191221
281216144441
37292916388817289261
4656134992699846220820736194481
559049393660104976013996809331202488324084101
6531441425152814171760251942402519424013436928298598485766121
7478296944641044178564176396809280529079040423263232188116992358318081801088541
843046721459165024214277011257140536329523422720101583175686772211712257989017642998169637822859361
938742048946490458682479491129677139724032154279448064205705930752182849716224104485552128348285173765159780352794280046581
10348678440146490458680278942752080991796451840231419172096037027067535364114118615040313456656384015672832819204643802316806191736422416679880978201
11313810596094602555409323068370272880122734810915203272928291072061094661433344814595485777927758052245504051720348303360229868214681606129819058176743008370688350277500542221
122824295364814518872583696331383989471041472817730982404418453192947209426033478287361466271874400256167573928502886413964494041907208275255728537603310102291415048024490403430489161004482567355827511386641
132541865828329440590076910363524720615282881723196745249408574398915083136013785573961995264245076870435471363267691605806284832676916058062848242051230059724801290939893651865646943268860067841043183752445952106993205379072154472377739119461
142287679245496142703345915927237009566460470241973843544558412872374263300475136192998035467933696385996070935867392588184489045131264686215237219319808609969099750506496406646066500337664197161729212284928657205764040949761348114387776307212839184645488643243919932521508681
Technical details
Node id
n-a88d7af0cffbf501 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c91d5d506c107735

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:         9,         12;
  2:        81,        216,         144;
  3:       729,       2916,        3888,        1728;
  4:      6561,      34992,       69984,       62208,        20736;
  5:     59049,     393660,     1049760,     1399680,       933120,       248832;
  6:    531441,    4251528,    14171760,    25194240,     25194240,     13436928,      2985984;
  7:   4782969,   44641044,   178564176,   396809280,    529079040,    423263232,    188116992,     35831808;
  8:  43046721,  459165024,  2142770112,  5714053632,   9523422720,  10158317568,   6772211712,   2579890176,   429981696;
  9: 387420489, 4649045868, 24794911296, 77139724032, 154279448064, 205705930752, 182849716224, 104485552128, 34828517376, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (9+12*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[(9 + 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; for(n = 1, N, my(i = n + 1); v[i] = (9 + 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

Proof chains ? JSON

Candidate equalities ?

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.