Real-rooted atlas

A038277 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(49 - 77\,x + 231\,n x + 121\,n^{2} x^{2}\right)\,P_{n}(x) = \left(343 + 1617\,x + 2541\,x^{2} + 1331\,x^{3} + 1617\,n x + 4235\,n x^{2} + 2662\,n x^{3} + 847\,n^{2} x^{2} + 1331\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
174451
24961610891754
33436468228692129650976
42401603683201665962883660251345248
51680752822037352701043504012810875579783633324048
611764944370483922033514609056026902837524350911286806489789209568
7823543362358923843592831789609360439413012559659732444253517961124717894418071828352
85764801289887136358735330820043624832615178217501113648338881190985029086984202086417363069361403112878848
94035360722828611963228617977221045806073677512455405017539961337362501068561068220002365721610938733697432357947691008804948500224
10282475249177555870802825040730052104580607360904311979725024555945872304437686998186905133388533504038285567941005165056338370003138428376721189036403720704
11197732674313671802051624169792912652025658834584099474317769750315134638694568674037977207826988177292699520982662910485795635466902724500241658985007517410848805679844000510168141824
121384128720110440285203042030262604662618906149122784010444803365823753781615664334816943653168090956416601378517351936206359211202016951779307327628600010149677370315714345112996771065653039439566584983634624316440576
13968890104077917216278972167958088203906172045957017334410560856736555125430158781818085321226749118518243322517875741798376963755737643876708494047924170355065003078735468995766581570264135308348484826589000559225967664515802104761730234136153083904
146782230728495968363041071213716577203318991532773071608979210349639601824022546839511797969290415027676701848480673525026038517727744613437148499862386779339313738959274007542901899039628121512952950867393836823650286102740206916631122548606266488544371255622922535470763817289408512
Technical details
Node id
n-9b87437013f6eb34 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e37eecb91b55dd80

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        7,         44;
  2:       49,        616,        1089;
  3:      343,       6468,       22869,        21296;
  4:     2401,      60368,      320166,       596288,       366025;
  5:    16807,     528220,     3735270,     10435040,     12810875,       5797836;
  6:   117649,    4437048,    39220335,    146090560,    269028375,     243509112,      86806489;
  7:   823543,   36235892,   384359283,   1789609360,   4394130125,    5965973244,    4253517961,    1247178944;
  8:  5764801,  289887136,  3587353308,  20043624832,  61517821750,  111364833888,  119098502908,   69842020864,   17363069361;
  9: 40353607, 2282861196, 32286179772, 210458060736, 775124554050, 1753996133736, 2501068561068, 2200023657216, 1093873369743, 235794769100;
%F Formula
The row polynomials satisfy (49+(231*n-77)*x+121*n^2*x^2)*P(n,x) = (343+(1617*n+1617)*x+(847*n^2+4235*n+2541)*x^2+(1331*n^2+2662*n+1331)*x^3)*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[((343 + (1617*n + 1617)*x + (847*n^2 + 4235*n + 2541)*x^2 + (1331*n^2 + 2662*n + 1331)*x^3)*P[n - 1, x])/(49 + (231*n - 77)*x + 121*n^2*x^2)]];
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] = ((343 + (1617*n + 1617)*x + (847*n^2 + 4235*n + 2541)*x^2 + (1331*n^2 + 2662*n + 1331)*x^3)*v[i - 1])/(49 + (231*n - 77)*x + 121*n^2*x^2)); 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.