Real-rooted atlas

A038327 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(13 + 27\,x + 14\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1131427
2169364196729
3219770987644274419683
42856112303219874414268838416531441
5371293199927043061204637360249704053782414348907
64826809311886128396934012057136097384560419502727529536387420489
762748517473027282152824198827429984402953998320190873737668518777610541350410460353203
881573072170278339042648952779257054367552768039563206616956236835629764352109630044161475789056282429536481
9106044993731027820708464427535359521112560167264179721257788819354596992641389560809728641335758336172667319552206610467847625597484987
10137858491849148462991222071947449592202066183167776038939605854240503219521808644516072631616027791216194560112233757708802685936081920289254654976205891132094649
11179216039403721230207744746114316503240940369330241239960795480519593760119933986031059212915967726421769935359789555205349809117452801920444298572804136341566156840495651696645559060566555523
12232980851224813010829461982161783337450558664640172418149264015511870132078320267281454583503363358151608869657630998322543412224208642555580659209986310352578560322634642160230463173216646758456693912375296150094635296999121
133028751065922534240251492291542273985481040376561081891386672256162912784502579151205646320728076508488107537455699601928731194183061109767052118378626280964219216123964441601817508484169297925338136806651084895812711914250247937147732541444052555153018976267
143937376385699289593635208920815884155446462445711161790038476130460192530126779469405518411418115250110272704184446477117165943682270110487595888537621391425748499718912153579466912305674248269663602970305536323846966270165811287189567841967718414445608873225420811112006825558016109418989131512359209
Technical details
Node id
n-0b1d1ff3e2f24ef7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b21306f08ea8390f

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,           14;
  2:         169,          364,          196;
  3:        2197,         7098,         7644,          2744;
  4:       28561,       123032,       198744,        142688,         38416;
  5:      371293,      1999270,      4306120,       4637360,       2497040,        537824;
  6:     4826809,     31188612,     83969340,     120571360,      97384560,      41950272,       7529536;
  7:    62748517,    473027282,   1528241988,    2742998440,    2953998320,    1908737376,     685187776,    105413504;
  8:   815730721,   7027833904,  26489527792,   57054367552,   76803956320,   66169562368,   35629764352,  10963004416,   1475789056;
  9: 10604499373, 102782070846, 442753535952, 1112560167264, 1797212577888, 1935459699264, 1389560809728, 641335758336, 172667319552, 20661046784;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (13+27*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[((13 + 27*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] = ((13 + 27*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.