Real-rooted atlas

A038303 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 25\,x + 14\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1111425
2121308196625
3133150826468274415625
4146417453614229612073638416390625
516105110248702608760332024021128805378249765625
6177156113528284430445407304528069725040354963847529536244140625
7194871711736129786628859161406121640178960936013666107845797742721054135046103515625
8214358881218256315297223267682474774086439371405920400872496642551006796892763883521475789056152587890625
9235794769127009219006137501478576408337724256779553837216992159429184841832242944459181223424146103116544206610467843814697265625
102593742460133011267674018906453304206416735666880142918203489602182750744204823150386680960168366448588808035671409920227271514624028925465497695367431640625
11285311670611399436338855425418676108980970531269615602470443231748804401880667479685602393576792325093085069811203241054135334401374992663475203499981325209640495651696642384185791015625
12313842837672147932360662648335526524638536142344586210288040762313323855208300689258675968123252658689431041344574458430156810695478646603520604996771929088023099876746383365345426023956485669391237529659604644775390625
1334522712143931571193964563222436184482030096820355275828071184647667867256810401483748204988329282517875741798376963204569125925207043058906892928606722162863459646489601101094124910940163821979607128883281072294696673287937147732541441490116119384765625
1437974983358324167664515802104765597700852719575628497386159299657699740851557548801625388580396466967684846910802961875648705005207703545548878511943585167572486661619455711187968423921238090711961619619495316594933766242566691643842561222320750811381761111200682555801637252902984619140625
Technical details
Node id
n-fc9820a414a31af3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-64aa42accc9ceb8f

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,          14;
  2:        121,         308,          196;
  3:       1331,        5082,         6468,         2744;
  4:      14641,       74536,       142296,       120736,        38416;
  5:     161051,     1024870,      2608760,      3320240,      2112880,       537824;
  6:    1771561,    13528284,     43044540,     73045280,     69725040,     35496384,      7529536;
  7:   19487171,   173612978,    662885916,   1406121640,   1789609360,   1366610784,    579774272,    105413504;
  8:  214358881,  2182563152,   9722326768,  24747740864,  39371405920,  40087249664,  25510067968,   9276388352,   1475789056;
  9: 2357947691, 27009219006, 137501478576, 408337724256, 779553837216, 992159429184, 841832242944, 459181223424, 146103116544, 20661046784;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+25*x+14*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 + 25*x + 14*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 + 25*x + 14*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.