Real-rooted atlas

A207824 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −2/3)coefficients of one signHadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(11 + 48\,x + 78\,x^{2} + 56\,x^{3} + 15\,x^{4} - 5\,n - 28\,n x - 56\,n x^{2} - 48\,n x^{3} - 15\,n x^{4}\right)\,P_{n-1}(x) + \left(8 + 46\,x + 105\,x^{2} + 119\,x^{3} + 67\,x^{4} + 15\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16814
2359463192
320482411084962632
4118964121295411620390536080
569304675212606016981611429030744494592
64039132716211034091983436200416910793542420476779968
7235416222548890112562025920027312520220803369911336190563292941184
81372105148283127007331618912783231887427034391104823170578089161032150030091274056704
97997214972503925253687441654846320334942610045174958724060138536234480379278958371011811844017465029120
104661117962990754238291135551378792509632568123910527295966125926239604645653447400230709017436906213270929944511239414180864
112716698604039089864272860749241105576248162985276567445640513765607609665927927330372760324941136598282219624039125980355263673214376483281938415616
12158340798125684768164190893628638859558288540261165385133556408757437848880920856052102690485624248655339974635518598223794020967072743185135909446845764155667344989481095168
139228778026162193443248131520399262865153298958162199946043503053466820958520962249340839281298426363847841313632735425189841538622512053070259529140195063129158164380130469578453811015736616724981669888
1453789260175101813929957089449249288054834658100194017959773771965548507286749636698230984597677315151835249618161788846374377845160863279494123010846206352509675317069659967561791497554236373713503579301035728465692158454191818997760
Technical details
Node id
n-970f28ded695053c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4ca9a374d9951802

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:       6,        8;
  2:      35,       94,        63;
  3:     204,      824,      1108,        496;
  4:    1189,     6412,     12954,      11620,       3905;
  5:    6930,    46752,    126060,     169816,     114290,      30744;
  6:   40391,   327162,   1103409,    1983436,    2004169,    1079354,     242047;
  7:  235416,  2225488,   9011256,   20259200,   27312520,   22080336,    9911336,    1905632;
  8: 1372105, 14828312,  70073316,  189127832,  318874270,  343911048,  231705780,   89161032,  15003009;
  9: 7997214, 97250392, 525368744, 1654846320, 3349426100, 4517495872, 4060138536, 2344803792, 789583710, 118118440;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(5*n-11+(28*n-48)*x+(56*n-78)*x^2+(48*n-56)*x^3+(15*n-15)*x^4)*P(n-1,x) + (8+46*x+105*x^2+119*x^3+67*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[(-(5*n - 11 + (28*n - 48)*x + (56*n - 78)*x^2 + (48*n - 56)*x^3 + (15*n - 15)*x^4)*P[n - 1, x] + (8 + 46*x + 105*x^2 + 119*x^3 + 67*x^4 + 15*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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] = (-(5*n - 11 + (28*n - 48)*x + (56*n - 78)*x^2 + (48*n - 56)*x^3 + (15*n - 15)*x^4)*v[i - 1] + (8 + 46*x + 105*x^2 + 119*x^3 + 67*x^4 + 15*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023