Real-rooted atlas

A038322 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(19 + 62\,x + 43\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1194362
2361163418493844
368594656910539379507238328
4130321117974840049346042532341880114776336
5247609928019015126822910287020270324786095147008443916132832
6470458816388335423614452935109067702601851280741516758962502632136304956800235584
7893871739141608101819614444807136265011114582073446206511144710063838407412855172718186111073521614606208
8169835630413074918782162435659351132110245633788083118790955847056466530990072638963376992924131642888826411688200277601218340105584896
93226876977796572638896867594996784347963142000562960281066626506899674241394419982557836420912488596443532554669946572199868224746977150259261193684313537086546263552
1061310662578011387557100449701413117362826405852828724232076033776506051823010917298795933719641729993343208330902237284624299495601898748135096282459549259626800017021611482313284249839299365868340224
1111649025889821928999943399398733281572542563540522280150420562985510084699664044298703195257472502456746723137217461082316211689812161964864510132279453411707677359978976310006017765451679980347640804192929373947122270752036560683837093888
12221331491906616160108973591481004748198539704487234564430477320928966028741394042526251295104074100532937162584274792142635211280156533055434585597821656753992884446733760895758402199560457350140514915177596310516674211878972599438777196399596307972625764013226266762397899821056
134205298346225705912372430397579839991680045811881894061813941432789826945460278879159205599822998532132878539544348947819696237032985312314076219441153904404436581723724724849166864778821346831335822858241371145423946829554295658728262616705311603068898370698700288069238563710471718264124282290785243200028539268669788905472
147990066857828841212531589604427874951837241015496715318369933712919291763340840122098185634868955291760194970507683542186890194322399881346761634443027833876384836736859010536165129468313064338527744324914270657760584409449142819246416536066130542929232014537628805442322193213127138313208728973492514570582570590893488746387388535734413850376544912401769434657526912139264
Technical details
Node id
n-291332f705353fde · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ee0c3b4e0db3251f

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:           19,            43;
  2:          361,          1634,           1849;
  3:         6859,         46569,         105393,           79507;
  4:       130321,       1179748,        4004934,         6042532,          3418801;
  5:      2476099,      28019015,      126822910,       287020270,        324786095,        147008443;
  6:     47045881,     638833542,     3614452935,     10906770260,      18512807415,      16758962502,       6321363049;
  7:    893871739,   14160810181,    96144448071,    362650111145,     820734462065,    1114471006383,     840741285517,     271818611107;
  8:  16983563041,  307491878216,  2435659351132,  11024563378808,   31187909558470,   56466530990072,   63896337699292,   41316428888264,   11688200277601;
  9: 322687697779, 6572638896867, 59499678434796, 314200056296028, 1066626506899674, 2413944199825578, 3642091248859644, 3532554669946572, 1998682247469771, 502592611936843;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (19+62*x+43*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[((19 + 62*x + 43*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] = ((19 + 62*x + 43*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.