Real-rooted atlas

A038220 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(13 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113821
216920864441
32197405624965129261
4285617030464896266244096194481
537129311424401406080865280266240327684084101
6482680917822064274185602249728010383360255590426214485766121
7627485172703013044990177925118131203149619201162936322385510420971521801088541
881573072140159050888649641728106457128968189009920403151257612404654082181038081677721637822859361
9106044993735873261191214457258316820759140147219162283212811792174284848378150912127590727681962934272134217728794280046581
101378584918498483599498402349304476480385526888448041518280294403065965314048157228990464055289315328012759072768017448304640107374182416679880978201
11179216039403712131547282712373278377929606891293131008084815915458560730721733181444496749127270419765930229760608182468608012475537817601535450808328589934592350277500542221
1223298085122481172047397827552582314269570176119449080937472016539103514419201628465576804352116915477309030461669702316851223719116275712064872796651520119765163048961340029796352687194767367355827511386641
133028751065922532423000852738024894646468703270420186894678432768310567610437427203440133530999193628226736664608768173702994859130888017061301190656274087565852672067467708517580811323251779174411613591568384549755813888154472377739119461
1439373763856992893392201193833233613568804775332934433400134831588761656523305099611750469567144737983692864215825911984947245162778663374028824318419280278323299767873970372608306978073754992646869439412699136105683683272294410005555812761643980465111043243919932521508681
Technical details
Node id
n-e8edca1dfa93add7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-0e62401a26557dd3

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:          13,           8;
  2:         169,         208,           64;
  3:        2197,        4056,         2496,          512;
  4:       28561,       70304,        64896,        26624,         4096;
  5:      371293,     1142440,      1406080,       865280,       266240,        32768;
  6:     4826809,    17822064,     27418560,     22497280,     10383360,      2555904,      262144;
  7:    62748517,   270301304,    499017792,    511813120,    314961920,    116293632,    23855104,     2097152;
  8:   815730721,  4015905088,   8649641728,  10645712896,   8189009920,   4031512576,  1240465408,   218103808,   16777216;
  9: 10604499373, 58732611912, 144572583168, 207591401472, 191622832128, 117921742848, 48378150912, 12759072768, 1962934272, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (13+8*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[(13 + 8*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] = (13 + 8*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

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.