Real-rooted atlas

A114656 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n}(x) = \left(1 - 20\,x - 75\,x^{2} - 86\,x^{3} - 32\,x^{4} + 13\,n + 64\,n x + 105\,n x^{2} + 70\,n x^{3} + 16\,n x^{4}\right)\,P_{n-1}(x) + \left(2 + 10\,x + 2\,x^{2} - 34\,x^{3} - 44\,x^{4} - 16\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2369
3114644101
4452885763601269
5197170052326800315217081
6903981040920818407848028896240849
742795605829562080252011824808969282738563511741
82079331894820351526997200139944001628121610206336266150452515549

All 15 rows

Technical details
Node id
n-eef72b68ff77fb86 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a22265fba64859ce

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:      3,        6;
   3:     11,       46,       44;
   4:     45,      288,      576,       360;
   5:    197,     1700,     5232,      6800,       3152;
   6:    903,     9810,    40920,     81840,      78480,      28896;
   7:   4279,    56058,   295620,    802520,    1182480,     896928,     273856;
   8:  20793,   318948,  2035152,   6997200,   13994400,   16281216,   10206336,    2661504;
   9: 103049,  1811464, 13570592,  56553280,  143334464,  226213120,  217129472,  115933696,   26380544;
  10: 518859, 10283166, 88480944, 433593216, 1333088064, 2666176128, 3468745728, 2831390208, 1316245248, 265655808;
%F Formula
The row polynomials satisfy (3*n+3+(6*n+6)*x+(3*n+3)*x^2)*P(n,x) = (13*n+1+(64*n-20)*x+(105*n-75)*x^2+(70*n-86)*x^3+(16*n-32)*x^4)*P(n-1,x) + (2+10*x+2*x^2-34*x^3-44*x^4-16*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[((13*n + 1 + (64*n - 20)*x + (105*n - 75)*x^2 + (70*n - 86)*x^3 + (16*n - 32)*x^4)*P[n - 1, x] + (2 + 10*x + 2*x^2 - 34*x^3 - 44*x^4 - 16*x^5)*D[P[n - 1, x], x])/(3*n + 3 + (6*n + 6)*x + (3*n + 3)*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] = ((13*n + 1 + (64*n - 20)*x + (105*n - 75)*x^2 + (70*n - 86)*x^3 + (16*n - 32)*x^4)*v[i - 1] + (2 + 10*x + 2*x^2 - 34*x^3 - 44*x^4 - 16*x^5)*deriv(v[i - 1], x))/(3*n + 3 + (6*n + 6)*x + (3*n + 3)*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

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