Real-rooted atlas

A207824 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, −4/5)coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(38 + 159\,x + 249\,x^{2} + 173\,x^{3} + 45\,x^{4} - 26\,n - 121\,n x - 209\,n x^{2} - 159\,n x^{3} - 45\,n x^{4}\right)\,P_{n-1}(x) + \left(32 + 172\,x + 369\,x^{2} + 395\,x^{3} + 211\,x^{4} + 45\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16713
2358248165
32047188423292093
41189558498317690225526549
5693040701955921122266586015456336765
640391284754836271130956411532765415561059374271749
723541619367006826980133671701570070011062866432974272610354185677

All 15 rows

Technical details
Node id
n-7d79207c469b3adb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4b698bc37dee3e22

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,        7;
  2:      35,       82,        48;
  3:     204,      718,       842,        329;
  4:    1189,     5584,      9831,       7690,       2255;
  5:    6930,    40701,     95592,     112226,      65860,      15456;
  6:   40391,   284754,    836271,    1309564,    1153276,     541556,     105937;
  7:  235416,  1936700,   6826980,   13367170,   15700700,   11062866,    4329742,    726103;
  8: 1372105, 12902576,  53072610,  124725020,  183165715,  172124424,  101076189,  33911382,   4976784;
  9: 7997214, 84612659, 397817896, 1090900560, 1922807410, 2259075113, 1769171928, 890553162, 261458430, 34111385;
%F Formula
The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(26*n-38+(121*n-159)*x+(209*n-249)*x^2+(159*n-173)*x^3+(45*n-45)*x^4)*P(n-1,x) + (32+172*x+369*x^2+395*x^3+211*x^4+45*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[(-(26*n - 38 + (121*n - 159)*x + (209*n - 249)*x^2 + (159*n - 173)*x^3 + (45*n - 45)*x^4)*P[n - 1, x] + (32 + 172*x + 369*x^2 + 395*x^3 + 211*x^4 + 45*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*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] = (-(26*n - 38 + (121*n - 159)*x + (209*n - 249)*x^2 + (159*n - 173)*x^3 + (45*n - 45)*x^4)*v[i - 1] + (32 + 172*x + 369*x^2 + 395*x^3 + 211*x^4 + 45*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*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