Real-rooted atlas

A088996 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 - 2\,x - x^{2} + 2\,n + 7\,n x + 5\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
23213660
3151474745041140
41051314611112534957629640
59451434686535259470387000229824978120
610395185211136791953636851178163013751064666489639124800
71351352761173240801211162371513355264805793392525541798242266064641838865600
820270254668786046888756226826819849566146089217756215003090692818825012110096883765209699298742400
93445942588288272010025354394662366464562806624908097910959934161483545125196178511071190412507729298563888566922246057223286400
10654729075184632835052337429867661749738647970857818300171528783285924545669510610561081066108146607801112351105505606867615071520019053977918976411891183475200
1113749310575423084323655590562003135649366632892734274627730026395106767088650805529602697221036985854184404906212809292131324388074488951347494404108645674919296102891480762470430891838760640000
12316234143225105421487679901607912358168331483864138382886922906913127839140759466461242570131078785912360299309302154916961058531534070094075748648806945580626040533983663473360768266075135395606656607059736498575362533130778372480000
137905853580625283792704366150469447841657218547387242551849462325694033138464479160964708593288127458855755013285319551612159617131508558633083689667037401412502383541255364988725487323610399408576040826832847185255552185464580065652643843885182313590882304225448639275150720000
1421345804667687582079069149491751463326138487133451603374559643416239120634586308161538116593335960674184344926997618746523917069184138780621717601571720056497568307507435583639210148415066166972494802380946985873768848885777780495938173443317805832633044771456138460552491995706470426807757963777087897621643069370414469120000
Technical details
Node id
n-2cc49758a2bbacea · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b903ea8a3904092b

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:        1,         4;
  2:        3,        21,          36;
  3:       15,       147,         474,         504;
  4:      105,      1314,        6111,       12534,         9576;
  5:      945,     14346,       86535,      259470,       387000,       229824;
  6:    10395,    185211,     1367919,     5363685,     11781630,     13751064,       6664896;
  7:   135135,   2761173,    24080121,   116237151,    335526480,    579339252,     554179824,     226606464;
  8:  2027025,  46687860,   468887562,  2682681984,   9566146089,  21775621500,   30906928188,   25012110096,    8837652096;
  9: 34459425, 882882720, 10025354394, 66236646456, 280662490809, 791095993416, 1483545125196, 1785110711904, 1250772929856, 388856692224;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(7*n-2)*x+(5*n-1)*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[((2*n - 1 + (7*n - 2)*x + (5*n - 1)*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] = ((2*n - 1 + (7*n - 2)*x + (5*n - 1)*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.