Real-rooted atlas

A147716 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(14 + 29\,x + 15\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1141529
2196420225841
3274488209450337524389
43841616464026460018900050625707281
5537824288120061740006615000354375075937520511149
67529536484041601296540001852200001488375006378750011390625594823321
71054135047906012802541218400453789000048620250003125587500111628125017085937517249876309

All 15 rows

Technical details
Node id
n-add9ebb7eb3bfbfb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-75a6722c02565e9c

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:          14,           15;
  2:         196,          420,          225;
  3:        2744,         8820,         9450,          3375;
  4:       38416,       164640,       264600,        189000,         50625;
  5:      537824,      2881200,      6174000,       6615000,       3543750,        759375;
  6:     7529536,     48404160,    129654000,     185220000,     148837500,      63787500,      11390625;
  7:   105413504,    790601280,   2541218400,    4537890000,    4862025000,    3125587500,    1116281250,     170859375;
  8:  1475789056,  12649620480,  47436076800,  101648736000,  136136700000,  116688600000,   62511750000,   19136250000,   2562890625;
  9: 20661046784, 199231522560, 853849382400, 2134623456000, 3430644840000, 3675690900000, 2625493500000, 1205583750000, 322924218750, 38443359375;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (14+29*x+15*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[((14 + 29*x + 15*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] = ((14 + 29*x + 15*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

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.