Real-rooted atlas

A207823 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(-5\,x - 36\,x^{2} - 78\,x^{3} - 68\,x^{4} - 21\,x^{5} + 2\,n + 21\,n x + 72\,n x^{2} + 110\,n x^{3} + 78\,n x^{4} + 21\,n x^{5}\right)\,P_{n-1}(x) + \left(-x - 13\,x^{2} - 54\,x^{3} - 94\,x^{4} - 73\,x^{5} - 21\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1156
21102435
311573115204
41201474705511189
51252461190282826406930
613037024008631163101264940391
7135519422520385582759137160605235416

All 15 rows

Technical details
Node id
n-43cc44645ac6de6e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4e678e331ffe3d5b

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: 1,  5;
  2: 1, 10,  24;
  3: 1, 15,  73,   115;
  4: 1, 20, 147,   470,   551;
  5: 1, 25, 246,  1190,  2828,   2640;
  6: 1, 30, 370,  2400,  8631,  16310,   12649;
  7: 1, 35, 519,  4225, 20385,  58275,   91371,   60605;
  8: 1, 40, 693,  6790, 41140, 157800,  374115,  501150,  290376;
  9: 1, 45, 892, 10220, 74571, 359275, 1142730, 2313450, 2704755, 1391275;
%F Formula
The row polynomials satisfy (2*n+6*n*x+6*n*x^2+2*n*x^3)*P(n,x) = (2*n+(21*n-5)*x+(72*n-36)*x^2+(110*n-78)*x^3+(78*n-68)*x^4+(21*n-21)*x^5)*P(n-1,x) - (x+13*x^2+54*x^3+94*x^4+73*x^5+21*x^6)*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[((2*n + (21*n - 5)*x + (72*n - 36)*x^2 + (110*n - 78)*x^3 + (78*n - 68)*x^4 + (21*n - 21)*x^5)*P[n - 1, x] - (x + 13*x^2 + 54*x^3 + 94*x^4 + 73*x^5 + 21*x^6)*D[P[n - 1, x], x])/(2*n + 6*n*x + 6*n*x^2 + 2*n*x^3)]];
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] = ((2*n + (21*n - 5)*x + (72*n - 36)*x^2 + (110*n - 78)*x^3 + (78*n - 68)*x^4 + (21*n - 21)*x^5)*v[i - 1] - (x + 13*x^2 + 54*x^3 + 94*x^4 + 73*x^5 + 21*x^6)*deriv(v[i - 1], x))/(2*n + 6*n*x + 6*n*x^2 + 2*n*x^3)); 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

Candidate equalities ?

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