Real-rooted atlas

A038277 → x ↦ x/(x+1) → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 18\,n x\right)\,P_{n}(x) = \left(49 + 252\,x + 324\,x^{2} + 126\,n x + 324\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
173643
2495049721525
33435292204122332849375
42401493922857686531845248801515625
516807432180333396011430720183708001133740844921875
61176493630312350065801600300803857868004761711362380855681298828125
782354329647548343064484196036848063011844001166619283211666192832489776025636865234375
8576480123718038432019351842195612697688216581600217768932864326653399296274274574336991796451841031494140625
94035360718677955242881741665623053933324811115289281603429860692608685972138521686396490915846248317646592198359290368028533935546875
10282475249145272985202521523957402305393332480129678374952004801804969651212004512424128020159181213696021869111763072013885150325760039275139492864782012939453125
1119773267431118601986042157303830220221894108251201426462124472006162316377719041848694913315712388064238363648056130720191884805345782875417600302418574095052877122092095078421266937255859375
1213841287201854205152976181213521738482071011677011201497785230695600739477965326284825881728786419968651947920450928641178745124029580801496819205116928001270158011199221766478255735986585615038807958540288574588775634765625
13968890104076477722410068149913004347288188462062608019215144272888144400841156185558648963364624742234595849887876793505751042145316125733837056340526369164101120038528126339709726722947606359873896448136853152422716620829152150811939635215437602996826171875
14678223072849488320612451281224289535502852167902564868962561484138743038151209159256242749733124121665309237379904138430275109080514563504016338698600524866743168356163819520943939095322888304649628847442254728396867058044687131144192285691077957008424965622200513731215360412762165069580078125
Technical details
Node id
n-ccf3ad4eb47077b4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-028fc00d5fe07c2e

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:        7,         36;
  2:       49,        504,         972;
  3:      343,       5292,       20412,        23328;
  4:     2401,      49392,      285768,       653184,        524880;
  5:    16807,     432180,     3333960,     11430720,      18370800,      11337408;
  6:   117649,    3630312,    35006580,    160030080,     385786800,     476171136,     238085568;
  7:   823543,   29647548,   343064484,   1960368480,    6301184400,   11666192832,   11666192832,    4897760256;
  8:  5764801,  237180384,  3201935184,  21956126976,   88216581600,  217768932864,  326653399296,  274274574336,   99179645184;
  9: 40353607, 1867795524, 28817416656, 230539333248, 1111528928160, 3429860692608, 6859721385216, 8639649091584, 6248317646592, 1983592903680;
%F Formula
The row polynomials satisfy (7+18*n*x)*P(n,x) = (49+(126*n+252)*x+(324*n+324)*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[((49 + (126*n + 252)*x + (324*n + 324)*x^2)*P[n - 1, x])/(7 + 18*n*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] = ((49 + (126*n + 252)*x + (324*n + 324)*x^2)*v[i - 1])/(7 + 18*n*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.