Real-rooted atlas

A119808 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 3\,x^{2} + 2\,x^{3} - 5\,n x - 16\,n x^{2} + 9\,n x^{3} - n^{2} x^{2} + 10\,n^{2} x^{3} + 3\,n^{3} x^{3}\right)\,P_{n}(x) = \left(2 - 6\,x^{2} + 4\,x^{3} - 8\,n x - 41\,n x^{2} - 20\,n x^{3} + 24\,n x^{4} - 3\,n^{2} x^{2} + 19\,n^{2} x^{3} + 20\,n^{2} x^{4} + 5\,n^{3} x^{3} + 4\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-x - 12\,x^{2} - 21\,x^{3} + 82\,x^{4} - 48\,x^{5} + 5\,n x^{3} + 35\,n x^{4} - 40\,n x^{5} + n^{2} x^{3} + 7\,n^{2} x^{4} - 8\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
241014
3834648
4169848162
53225823416540
6646429001761782
7128153830061120405832
825635869144540856018954
95128194260342200044009661236
101024184347057279568256001632196830
1120484096218423026412812300015264224629856
1240969011446684882144051672010396844802007666
138192196610115497024273441964480576576486085126377292
141638442598628015566884080691360027619203794561177620194758
Technical details
Node id
n-5bb174e91e3ec5cd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-661436938be51f0e

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2,    2;
  2:   4,   10;
  3:   8,   34,     6;
  4:  16,   98,    48;
  5:  32,  258,   234,    16;
  6:  64,  642,   900,   176;
  7: 128, 1538,  3006,  1120,   40;
  8: 256, 3586,  9144,  5408,  560;
  9: 512, 8194, 26034, 22000, 4400, 96;
%F Formula
The row polynomials satisfy (1-5*n*x-(n^2+16*n+3)*x^2+(3*n^3+10*n^2+9*n+2)*x^3)*P(n,x) = (2-8*n*x-(3*n^2+41*n+6)*x^2+(5*n^3+19*n^2-20*n+4)*x^3+(4*n^3+20*n^2+24*n)*x^4)*P(n-1,x) - (x+12*x^2-(n^2+5*n-21)*x^3-(7*n^2+35*n+82)*x^4+(8*n^2+40*n+48)*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[((2 - 8*n*x - (3*n^2 + 41*n + 6)*x^2 + (5*n^3 + 19*n^2 - 20*n + 4)*x^3 + (4*n^3 + 20*n^2 + 24*n)*x^4)*P[n - 1, x] - (x + 12*x^2 - (n^2 + 5*n - 21)*x^3 - (7*n^2 + 35*n + 82)*x^4 + (8*n^2 + 40*n + 48)*x^5)*D[P[n - 1, x], x])/(1 - 5*n*x - (n^2 + 16*n + 3)*x^2 + (3*n^3 + 10*n^2 + 9*n + 2)*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 - 8*n*x - (3*n^2 + 41*n + 6)*x^2 + (5*n^3 + 19*n^2 - 20*n + 4)*x^3 + (4*n^3 + 20*n^2 + 24*n)*x^4)*v[i - 1] - (x + 12*x^2 - (n^2 + 5*n - 21)*x^3 - (7*n^2 + 35*n + 82)*x^4 + (8*n^2 + 40*n + 48)*x^5)*deriv(v[i - 1], x))/(1 - 5*n*x - (n^2 + 16*n + 3)*x^2 + (3*n^3 + 10*n^2 + 9*n + 2)*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

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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.