Real-rooted atlas

A114656 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-n - 2\,n x - n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-2\,x - 6\,x^{2} - 6\,x^{3} - 2\,x^{4} - 2\,n - n x + 9\,n x^{2} + 13\,n x^{3} + 5\,n x^{4} + 2\,n^{2} + 5\,n^{2} x + 2\,n^{2} x^{2} - 3\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 7\,x^{2} - 9\,x^{3} - 5\,x^{4} - x^{5} + 4\,n x + 14\,n x^{2} + 18\,n x^{3} + 10\,n x^{4} + 2\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2246
34201943
487215696332
51622481611045012661
632640344076007430266821810
76417281272040480635404851614407181455
8128448043008183680409360496272310520785921526040

All 15 rows

Technical details
Node id
n-b455bb8d8c0e1d5c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3ff6c47fdabd4f1c

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,     4;
   3:   4,    20,     19;
   4:   8,    72,    156,      96;
   5:  16,   224,    816,    1104,      501;
   6:  32,   640,   3440,    7600,     7430,     2668;
   7:  64,  1728,  12720,   40480,    63540,    48516,    14407;
   8: 128,  4480,  43008,  183680,   409360,   496272,   310520,    78592;
   9: 256, 11264, 136192,  745472,  2201920,  3753344,  3699136,  1959488,   432073;
  10: 512, 27648, 410112, 2784768, 10426752, 23353344, 32133696, 26650944, 12235626, 2390004;
%F Formula
The row polynomials satisfy (n^2-n+(2*n^2-2*n)*x+(n^2-n)*x^2)*P(n,x) = (2*n^2-2*n+(5*n^2-n-2)*x+(2*n^2+9*n-6)*x^2-(3*n^2-13*n+6)*x^3-(2*n^2-5*n+2)*x^4)*P(n-1,x) + ((4*n-2)*x+(14*n-7)*x^2+(18*n-9)*x^3+(10*n-5)*x^4+(2*n-1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 2*n + (5*n^2 - n - 2)*x + (2*n^2 + 9*n - 6)*x^2 - (3*n^2 - 13*n + 6)*x^3 - (2*n^2 - 5*n + 2)*x^4)*P[n - 1, x] + ((4*n - 2)*x + (14*n - 7)*x^2 + (18*n - 9)*x^3 + (10*n - 5)*x^4 + (2*n - 1)*x^5)*D[P[n - 1, x], x])/(n^2 - n + (2*n^2 - 2*n)*x + (n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 2*n + (5*n^2 - n - 2)*x + (2*n^2 + 9*n - 6)*x^2 - (3*n^2 - 13*n + 6)*x^3 - (2*n^2 - 5*n + 2)*x^4)*v[i - 1] + ((4*n - 2)*x + (14*n - 7)*x^2 + (18*n - 9)*x^3 + (10*n - 5)*x^4 + (2*n - 1)*x^5)*deriv(v[i - 1], x))/(n^2 - n + (2*n^2 - 2*n)*x + (n^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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023