Real-rooted atlas

A038271 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(12 + 29\,x + 17\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1121729
2144408289841
31728734410404491324389
42073611750424969623582483521707281
52488321762560499392070747205011260141985720511149
62985984253808648989056016979328018040536010222970424137569594823321
7358318083553320961510161408356565888050513500804293647568202755579641033867317249876309
84299816964873125888241625825286846065049612123240192013739672217697322678208393925126086975757441500246412961
95159780352657871994883727941304321232291708928261861988147237097114987523503616415488212719568083275338180362811858787649714507145975869
106191736422487716265984055919119564802112500072448052372397629440890330759700481051084924646408508782723328045202908217680142305451796402015993900449420707233300201
11743008370688115785471098888201470869504034856251195392098759378386944019587276713410562774864201066496280789829869824019889279615779209392159818562402661111948592683427189630763312200509765705829
12891610044825615157370762035211810118052085765577000191262720177766881096499204029382638187315266596740825595904808674710025093127160140661680512045082367129099520191600060298672964935153068299152582622237229761353814783205469041
1310699320537907219704581990645761674889469204889687001202983698432308129260567265280785729614446526464148415593839899443221025542460652421122233963886444319744175821231803488128099632031355309939238494193932733385690889069007842716990457803290593710260628712958602189
14128391846454886425464382880219136234484525688684544132874564556921241651765715775300567041466695280300182732831167274706378883072504613019055658106886255098882044095283259075933885972011008418454531692301744642155674860233069593676346817966587881441663969109528197416168377826559400929297558232675799463481
Technical details
Node id
n-48ee1acaa798d2f7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8b732aeffa021ba5

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:         12,          17;
  2:        144,         408,          289;
  3:       1728,        7344,        10404,          4913;
  4:      20736,      117504,       249696,        235824,         83521;
  5:     248832,     1762560,      4993920,       7074720,       5011260,       1419857;
  6:    2985984,    25380864,     89890560,     169793280,     180405360,     102229704,      24137569;
  7:   35831808,   355332096,   1510161408,    3565658880,    5051350080,    4293647568,    2027555796,     410338673;
  8:  429981696,  4873125888,  24162582528,   68460650496,  121232401920,  137396722176,   97322678208,   39392512608,   6975757441;
  9: 5159780352, 65787199488, 372794130432, 1232291708928, 2618619881472, 3709711498752, 3503616415488, 2127195680832, 753381803628, 118587876497;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (12+29*x+17*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[((12 + 29*x + 17*x^2)*P[n - 1, x])/(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] = ((12 + 29*x + 17*x^2)*v[i - 1])/(1 + 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.