Real-rooted atlas

A125662 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-8 - 13\,x - 5\,x^{2} + 16\,n + 19\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-12 - 28\,x - 21\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
21522845
3561228821287
4209604651310551829
5780280940282874102014411655
6291112554224712136411376321637774269
7108645458011712013913098800419349848987473263
8405452325365817308290207360154168041470092952225843015765
91513169754712787384463332049369403496597164593249649887080676519217303
10564719404220612988602246704163067193526079354153568876183444162917725361017711122458061
11210756016584878591913841264639111797080241783294241260895446351947822341032522454873106446368780337215
1278655216749054826488243762873997010052224601140348128941158918569369972250569610782258801644379220896321213934972528261
1329354524272758757116753269230481756345415282520691285050965232236404606928652243491882495118626026693161650956860593066831781131686348967
14109552575109587701050806779851446930522028278072145401177165784260436845334405349592212299927809960766440349782008289141919215584880316730830832040201914329725
Technical details
Node id
n-e4bf922e77875fba · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-63c829d69f89a5ff

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:      4,      3;
  2:     15,     22,       8;
  3:     56,    122,      88,      21;
  4:    209,    604,     651,     310,      55;
  5:    780,   2809,    4028,    2874,    1020,     144;
  6:   2911,  12554,   22471,   21364,   11376,    3216,     377;
  7:  10864,  54580,  117120,  139130,   98800,   41934,    9848,    987;
  8:  40545, 232536,  581730,  829020,  736015,  416804,  147009,  29522,  2584;
  9: 151316, 975471, 2787384, 4633320, 4936940, 3496597, 1645932, 496498, 87080, 6765;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (16*n-8+(19*n-13)*x+(5*n-5)*x^2)*P(n-1,x) - (12+28*x+21*x^2+5*x^3)*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[((16*n - 8 + (19*n - 13)*x + (5*n - 5)*x^2)*P[n - 1, x] - (12 + 28*x + 21*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n)]];
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] = ((16*n - 8 + (19*n - 13)*x + (5*n - 5)*x^2)*v[i - 1] - (12 + 28*x + 21*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n)); 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

Same fingerprint ?

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