Real-rooted atlas

A038259 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 42\,x + 31\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1113142
21216829611764
3133111253317132979174088
41464116504469768613108049235213111696
5161051226935512790910360471105079365528629151130691232
6177156129955486211050015793036420167619061518895239668875036815489031744
71948717138442873732501702311526595108543022225785727466726916833778343727512614111230539333248
82143588814832818408476691633882686807390969464889672702133902398936300686247122824211100417688528910374419682651996416
923579476915980612779967417816791644332321950841874048155194652814084373666992264615505241198449470675168443621270665926439622160671406671383849472
10259374246017309637842109269949808845696650773513203435754951190101161909856220652272872769263941043943147258089204643991698866245290835843767381081962828698080117080198121677824
1128531167061188446617889411246293252078051053684294938715593894784420003023431848767116482660352101618737221329280204557198301873076651876052151759556854792655059917502272467692125408476896404831717368321110468608
1231384283767211061359414672921645107092743026154540363257678209799263942930049544185771960848223214527746235612218843509299740031003512618115295119097209577420501610876822206545551499828676786335391895032543769278766278378854976130129469486639681536
1334522712143931126478663581856321386392205659338220992719458479826155699415982110786578982067380016198972967811016703638991683638310470738917036176780974404061801917276778293258884639365312004621491833593466239805204948268794978112635778081762615823244175462974450425911265437718438866624512
143797498335832411498285707046605427445869997262817130938980724187175642397771006124506112113514709307247216268257130362071545050588318400428303562561747924537378343037586249203852477143237364689244212012177927435593348441123100005206777981422048672954912295721418371376030212980653655901475694393522079632032153148384174432398229504
Technical details
Node id
n-aad6f7f53daa6d5b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5e64a37ca1ebb259

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,          31;
  2:        121,         682,          961;
  3:       1331,       11253,        31713,         29791;
  4:      14641,      165044,       697686,       1310804,         923521;
  5:     161051,     2269355,     12790910,      36047110,       50793655,       28629151;
  6:    1771561,    29955486,    211050015,     793036420,     1676190615,     1889523966,      887503681;
  7:   19487171,   384428737,   3250170231,   15265951085,    43022225785,    72746672691,    68337783437,     27512614111;
  8:  214358881,  4832818408,  47669163388,  268680739096,   946488967270,  2133902398936,  3006862471228,   2421110041768,   852891037441;
  9: 2357947691, 59806127799, 674178167916, 4433232195084, 18740481551946, 52814084373666, 99226461550524, 119844947067516, 84436212706659, 26439622160671;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+42*x+31*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[((11 + 42*x + 31*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] = ((11 + 42*x + 31*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.