Real-rooted atlas

A106800 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-4 - 4\,x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
211
3314
4117119
54946131109
6257322139231742
71539242714293784115815
8102991964514744543899175151193

All 15 rows

Technical details
Node id
n-6850f58687c014b0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-75d63132b26da3ac

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:      1;
   3:      3,       1;
   4:     11,       7,       1;
   5:     49,      46,      13,      1;
   6:    257,     322,     139,     23,      1;
   7:   1539,    2427,    1429,    378,     41,     1;
   8:  10299,   19645,   14744,   5438,    991,    75,    1;
   9:  75905,  170236,  155822,  74178,  19261,  2574,  141,   1;
  10: 609441, 1573732, 1703914, 997646, 338671, 65938, 6707, 271, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-3+(n-2)*x)*P(n-1,x) - (4+4*x+x^2)*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[(2*n - 3 + (n - 2)*x)*P[n - 1, x] - (4 + 4*x + x^2)*D[P[n - 1, x], x]];
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 - 3 + (n - 2)*x)*v[i - 1] - (4 + 4*x + x^2)*deriv(v[i - 1], x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.