Real-rooted atlas

A094816 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 6\,x + 6\,x^{2} + 2\,x^{3} - n - 3\,n x - 3\,n x^{2} - n x^{3}\right)\,P_{n-2}(x) + \left(3\,x + 3\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2257
35252959
416117267193593
5656082019285214577001
632636371551831907317981234195527
71957250291284963410394957563755371161251483939
81370019602511645793742288705069778105484721027120332925902193
910960117260241156993443294120992194191429466801266751346322777613627073502395761
1098641016897653125575020533210574142941893425140854072905915620213140705490121033816752531710726232327
119864101182176889148170811570099264862141215797344204687811625561763015999257375837352192643136453859212222710781250069560779
1210850511221453794531890958335598332156887335880641946792577284837132010937909715536802926011267415517426683013990227218990708467316654085456322828847953
13130206134527405058680259735468929146916065755655337119458301464316674134427946503676797387971881134843891645772709427526064357152130395806257773716502764244482196718129172358975916361
14169267974863774571882613822476394090233241917409779583024395117228032538875053460129115098940295841348006674111360439803278449896863785485666162680123562982598719795491536654344273927078173593056535039632468705447
152369751648055576420820853600253558808063924589747050531743776117116843557648845560182613249259123981547237781092101008653241415695691783133420057141106970256672688273453831424572144680230954034350956326821254190708285129134443910166077157345204719628979
Technical details
Node id
n-56a7795ad4f51cff · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-22586c14a2a87f68

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,        5;
   3:      5,       25,        29;
   4:     16,      117,       267,       193;
   5:     65,      608,      2019,      2852,       1457;
   6:    326,     3637,     15518,     31907,      31798,      12341;
   7:   1957,    25029,    128496,    341039,     495756,     375537,     116125;
   8:  13700,   196025,   1164579,   3742288,    7050697,    7810548,    4721027,    1203329;
   9: 109601,  1726024,  11569934,  43294120,   99219419,  142946680,  126675134,   63227776,  13627073;
  10: 986410, 16897653, 125575020, 533210574, 1429418934, 2514085407, 2905915620, 2131407054, 901210338, 167525317;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n-2,x) + (n+(2*n+3)*x+(n+3)*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 5*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*P[n - 2, x] + (n + (2*n + 3)*x + (n + 3)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 5*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*v[i - 2] + (n + (2*n + 3)*x + (n + 3)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023