Real-rooted atlas

A147716 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118321
23241089441
358322916486279261
41049766998417496194481194481
5188956815746405248808748072902434084101
634012224340122241417176031492803936602624472985766121
7612220032714256704357128352992023201653372016533729185421871801088541
8110199605761469328076885710804482857026816595213920793618566613488314928656137822859361
91983592903682975389355521983592903687713972403219284931008321415516835712835225509168106288219683794280046581
1035704672266245950778711040446308403328019835929036805785479302401157095860481607077584015305500809565938035429405904916679880978201
1164268410079232117825418478592981878487321604909392436608016364641455360381841633958463640272326475762228960631351908035075106011691702177147350277500542221
121156831381426176231366276285235221208575326146561178254184785920441845319294720117825418478592229104980375043272928291072340930030320252540763201262703816382637525314417355827511386641
13208229648656711684511642387562086445116423875620864275711479239905281148797830166272034463934904988167658652201108481276442033518081595552541897614773634647209849089764844768589841243571941594323154472377739119461
14374813367582081024874564524358189056947444901388038144631629934258692096289497053201900544964990177339668482412475443349171245951913206650886701320675969927445911862188862049265518243760561546561566900644440176939647829693243919932521508681
Technical details
Node id
n-d19e6641dcda14df · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a5035a1c3b326aa3

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:           18,            3;
  2:          324,          108,            9;
  3:         5832,         2916,          486,          27;
  4:       104976,        69984,        17496,        1944,          81;
  5:      1889568,      1574640,       524880,       87480,        7290,        243;
  6:     34012224,     34012224,     14171760,     3149280,      393660,      26244,       729;
  7:    612220032,    714256704,    357128352,    99202320,    16533720,    1653372,     91854,     2187;
  8:  11019960576,  14693280768,   8571080448,  2857026816,   595213920,   79361856,   6613488,   314928,    6561;
  9: 198359290368, 297538935552, 198359290368, 77139724032, 19284931008, 3214155168, 357128352, 25509168, 1062882, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (18+3*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[(18 + 3*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] = (18 + 3*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.