Real-rooted atlas

A055151 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - 6\,x + 8\,x^{2} - n + 5\,n x + 4\,n x^{2} + 3\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-2 - 2\,x - 12\,x^{2} - 16\,x^{3} - 11\,n - 19\,n x + 8\,n x^{2} + 16\,n x^{3} + 5\,n^{2} + 20\,n^{2} x + 16\,n^{2} x^{2} + 5\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-12 - 58\,x - 110\,x^{2} - 96\,x^{3} - 32\,x^{4} - 18\,n - 66\,n x - 80\,n x^{2} - 32\,n x^{3} - 6\,n^{2} - 14\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2314
37310
41914235
5515010111
6141180605386
7393616280351324
8110720721232252144677
9313968405040142812616573
1089532230519740735010504259440
11256537199574580346506930462214270
1273789230538274230153780405904356132777415
132129417333829864146503642153583260417162832779
1461622723204093485482264864610630622132131801842910365486
Technical details
Node id
n-3b7d609dbe56e329 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4749f5f4c0c52e68

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:    3,    1;
  3:    7,    3;
  4:   19,   14,    2;
  5:   51,   50,   10;
  6:  141,  180,   60,    5;
  7:  393,  616,  280,   35;
  8: 1107, 2072, 1232,  252,  14;
  9: 3139, 6840, 5040, 1428, 126;
%F Formula
The row polynomials satisfy (n^3+3*n^2-n-6+(4*n^2+5*n-6)*x+(4*n+8)*x^2)*P(n,x) = (5*n^3+5*n^2-11*n-2+(4*n^3+20*n^2-19*n-2)*x+(16*n^2+8*n-12)*x^2+(16*n-16)*x^3)*P(n-1,x) - (6*n^2+18*n+12+(14*n^2+66*n+58)*x+(8*n^2+80*n+110)*x^2+(32*n+96)*x^3+32*x^4)*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[((5*n^3 + 5*n^2 - 11*n - 2 + (4*n^3 + 20*n^2 - 19*n - 2)*x + (16*n^2 + 8*n - 12)*x^2 + (16*n - 16)*x^3)*P[n - 1, x] - (6*n^2 + 18*n + 12 + (14*n^2 + 66*n + 58)*x + (8*n^2 + 80*n + 110)*x^2 + (32*n + 96)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 - n - 6 + (4*n^2 + 5*n - 6)*x + (4*n + 8)*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] = ((5*n^3 + 5*n^2 - 11*n - 2 + (4*n^3 + 20*n^2 - 19*n - 2)*x + (16*n^2 + 8*n - 12)*x^2 + (16*n - 16)*x^3)*v[i - 1] - (6*n^2 + 18*n + 12 + (14*n^2 + 66*n + 58)*x + (8*n^2 + 80*n + 110)*x^2 + (32*n + 96)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 - n - 6 + (4*n^2 + 5*n - 6)*x + (4*n + 8)*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. [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