Real-rooted atlas

A106800 → θ+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(1 + 6\,x + 6\,x^{2} - 4\,n - 14\,n x - 11\,n x^{2} + 4\,n^{2} + 10\,n^{2} x + 6\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3} - 2\,n - 7\,n x - 8\,n x^{2} - 3\,n x^{3} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(5 + 8\,x - 6\,n - 8\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2 + 4\,x + 2\,x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
211
3325
41012325
53866334141
6161370285805901
774721502265104018566393
837531303417535113403500420749589

All 15 rows

Technical details
Node id
n-578b7b68dbbf1b53 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a464fc6e981d0b75

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,      2;
   4:     10,     12,       3;
   5:     38,     66,      33,       4;
   6:    161,    370,     285,      80,      5;
   7:    747,   2150,    2265,    1040,    185,      6;
   8:   3753,  13034,   17535,   11340,   3500,    420,     7;
   9:  20253,  82642,  135891,  114240,  51100,  11298,   945,    8;
  10: 116642, 548118, 1068585, 1111208, 657510, 217014, 35707, 2112, 9;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(n^3-4*n^2+4*n-1+(2*n^3-10*n^2+14*n-6)*x+(n^3-6*n^2+11*n-6)*x^2)*P(n-2,x) + (n^2-2*n+1+(3*n^2-7*n+4)*x+(3*n^2-8*n+5)*x^2+(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+5+(2*n^2-8*n+8)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 4*n - 1 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*P[n - 2, x] + (n^2 - 2*n + 1 + (3*n^2 - 7*n + 4)*x + (3*n^2 - 8*n + 5)*x^2 + (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 5 + (2*n^2 - 8*n + 8)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n^3 - 4*n^2 + 4*n - 1 + (2*n^3 - 10*n^2 + 14*n - 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*v[i - 2] + (n^2 - 2*n + 1 + (3*n^2 - 7*n + 4)*x + (3*n^2 - 8*n + 5)*x^2 + (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 5 + (2*n^2 - 8*n + 8)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 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. [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.