Real-rooted atlas

A114687 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(1 - 4\,x - 26\,x^{2} - 36\,x^{3} - 15\,x^{4} + 8\,n + 43\,n x + 77\,n x^{2} + 57\,n x^{3} + 15\,n x^{4}\right)\,P_{n-1}(x) + \left(-1 - 11\,x - 42\,x^{2} - 70\,x^{3} - 53\,x^{4} - 15\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13710
2115055116
3453036634691480
4197175657508172423720112
59031001543670933909771539907285600
6427956766309165883540139510511518863877394188480
72079332105520971657504315158679151980111713482735385850562953600
81030491815128138274365945223215765597426377988027164628415725074439130777964661504
951885910266903893595964487097721431528714300681750641546389083638412540183087053140297203115014095360
1026467235812305056899123532680527601218882231030827034588535072353906289483452047880527415213018348904202705311236700808192
111364886932938907135821696352316257278598896851570292620560694611926268646903912006690923770389465621682431755247812865223442994267173772008581120
1271039373186873138022351823010160682227860772905619035261963633808864122563386201141610170932014664840932115132491959175407987415272386288352901706047118969392560662044659712
1337269351910613543343138472033986109554180364258622249575892239612430870162825159556588130970428128540202824184832841230356904504441186424793496370101760898025930335662372439555051001609115983282956738560
141968801519603441517988528137300417363751010252433879845188791844837346452105835684833468491394659386315624252962384538563734640911430825703524318967364195258222454201166012875519211077253909363568600705451354225779516047638784589824
Technical details
Node id
n-de5608bc74fa9bc0 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-37e408febc5a0069

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,        7;
  2:     11,       50,       55;
  3:     45,      303,      663,       469;
  4:    197,     1756,     5750,      8172,       4237;
  5:    903,    10015,    43670,     93390,      97715,      39907;
  6:   4279,    56766,   309165,    883540,    1395105,    1151886,     387739;
  7:  20793,   321055,  2097165,   7504315,   15867915,   19801117,   13482735,    3858505;
  8: 103049,  1815128, 13827436,  59452232,  157655974,  263779880,  271646284,  157250744,   39130777;
  9: 518859, 10266903, 89359596, 448709772, 1431528714, 3006817506, 4154638908, 3638412540, 1830870531, 402972031;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (8*n+1+(43*n-4)*x+(77*n-26)*x^2+(57*n-36)*x^3+(15*n-15)*x^4)*P(n-1,x) - (1+11*x+42*x^2+70*x^3+53*x^4+15*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[((8*n + 1 + (43*n - 4)*x + (77*n - 26)*x^2 + (57*n - 36)*x^3 + (15*n - 15)*x^4)*P[n - 1, x] - (1 + 11*x + 42*x^2 + 70*x^3 + 53*x^4 + 15*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] = ((8*n + 1 + (43*n - 4)*x + (77*n - 26)*x^2 + (57*n - 36)*x^3 + (15*n - 15)*x^4)*v[i - 1] - (1 + 11*x + 42*x^2 + 70*x^3 + 53*x^4 + 15*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