Real-rooted atlas

A133081 → reverse (shift +1) → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-3\,x^{2} - 6\,n x - 3\,n x^{2} + 10\,n^{2} x + 14\,n^{2} x^{2} - 4\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-x^{3} - 2\,n x^{2} - 2\,n x^{3} + 2\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
10235
2000055
3000006713
40000000099
5000000000101121
60000000000001313
70000000000000141529
800000000000000001717
900000000000000000181937
10000000000000000000002121
11000000000000000000000222345
120000000000000000000000002525
130000000000000000000000000262753
1400000000000000000000000000002929
Technical details
Node id
n-5dd4c3d4c56697ae · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3143880379ddecce

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0, 2, 3;
  2: 0, 0, 0, 0, 5;
  3: 0, 0, 0, 0, 0, 6, 7;
  4: 0, 0, 0, 0, 0, 0, 0, 0, 9;
  5: 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11;
  6: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0, 0, 13;
  7: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0, 0,  0, 14, 15;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0, 0,  0,  0,  0, 0, 17;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0, 0,  0,  0,  0, 0,  0, 18, 19;
%F Formula
The row polynomials satisfy (2*n^2-3*n+1)*P(n,x) = -((4*n^3-10*n^2+6*n)*x+(8*n^3-14*n^2+3*n+3)*x^2)*P(n-1,x) + ((2*n^2-2*n)*x^2+(4*n^2-2*n-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^3 - 10*n^2 + 6*n)*x + (8*n^3 - 14*n^2 + 3*n + 3)*x^2)*P[n - 1, x] + ((2*n^2 - 2*n)*x^2 + (4*n^2 - 2*n - 1)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^3 - 10*n^2 + 6*n)*x + (8*n^3 - 14*n^2 + 3*n + 3)*x^2)*v[i - 1] + ((2*n^2 - 2*n)*x^2 + (4*n^2 - 2*n - 1)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1)); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.