Real-rooted atlas

A114687 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-8\,x - 28\,x^{2} - 32\,x^{3} - 12\,x^{4} + 9\,n + 41\,n x + 67\,n x^{2} + 47\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}(x) + \left(-2 - 16\,x - 48\,x^{2} - 68\,x^{3} - 46\,x^{4} - 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
211362976
345219351185800
419712723050321612578992
590372652319036690287558925105728
6427941220164295346680408285254340654451284544
7207932333031115037294171546253554332965223849549182515999488
81030491319760735467623288160458169745733672044560292196583043768209203206144
9518859746831747543436175660548415041354650211030675273564448217748172504323293244052621763584
10264672342295140302800305127875240035271933906638845752863383269076594989604435482495151371546023115313334266216448
1113648869239763315190668703590595838852857414725062807196606981601998301090675848908442291874543349519455132878454391841801065452730896384
127103937313606019281189911670262826168240223023929535560706399792102361331701213670543335681325435738715909742549320419551261998116714415888148100694976036808941568
1337269351977292795457372616214642822729807016896912665494782399256635998813294622015583978411716181630258099291561633517801596267067739704028847564742102589098600312002965780581135093284864
1419688015194395368653245411216041128776254204881249400934681939314891099484924549741545431650543253238882247647784031652323189940711641794001264665611003562535153363844072872958590241250699889794701406611097980336390144
Technical details
Node id
n-5fd44c17d91e2308 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-06597c4b2866cc52

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:      3,       5;
  2:     11,      36,       29;
  3:     45,     219,      351,       185;
  4:    197,    1272,     3050,      3216,      1257;
  5:    903,    7265,    23190,     36690,     28755,      8925;
  6:   4279,   41220,   164295,    346680,    408285,    254340,     65445;
  7:  20793,  233303,  1115037,   2941715,   4625355,   4332965,   2238495,    491825;
  8: 103049, 1319760,  7354676,  23288160,  45816974,  57336720,  44560292,  19658304,   3768209;
  9: 518859, 7468317, 47543436, 175660548, 415041354, 650211030, 675273564, 448217748, 172504323, 29324405;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (9*n+(41*n-8)*x+(67*n-28)*x^2+(47*n-32)*x^3+(12*n-12)*x^4)*P(n-1,x) - (2+16*x+48*x^2+68*x^3+46*x^4+12*x^5)*d/dx 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[((9*n + (41*n - 8)*x + (67*n - 28)*x^2 + (47*n - 32)*x^3 + (12*n - 12)*x^4)*P[n - 1, x] - (2 + 16*x + 48*x^2 + 68*x^3 + 46*x^4 + 12*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (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] = ((9*n + (41*n - 8)*x + (67*n - 28)*x^2 + (47*n - 32)*x^3 + (12*n - 12)*x^4)*v[i - 1] - (2 + 16*x + 48*x^2 + 68*x^3 + 46*x^4 + 12*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023