Real-rooted atlas

A133081 → reverse (shift +1) → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(3\,x + 19\,x^{2} + 45\,x^{3} + 51\,x^{4} + 28\,x^{5} + 6\,x^{6} - 2\,n x - 14\,n x^{2} - 36\,n x^{3} - 44\,n x^{4} - 26\,n x^{5} - 6\,n x^{6}\right)\,P_{n-1}(x) + \left(x^{2} + 7\,x^{3} + 18\,x^{4} + 22\,x^{5} + 13\,x^{6} + 3\,x^{7}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
10123
2000011
300000123
40000000011
5000000000123
600000000000011
70000000000000123

All 15 rows

Technical details
Node id
n-bb1bd37fa8f0139d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-538c820bb67d4056

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, 1, 2;
  2: 0, 0, 0, 0, 1;
  3: 0, 0, 0, 0, 0, 1, 2;
  4: 0, 0, 0, 0, 0, 0, 0, 0, 1;
  5: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2;
  6: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  7: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -((2*n-3)*x+(14*n-19)*x^2+(36*n-45)*x^3+(44*n-51)*x^4+(26*n-28)*x^5+(6*n-6)*x^6)*P(n-1,x) + (x^2+7*x^3+18*x^4+22*x^5+13*x^6+3*x^7)*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*n - 3)*x + (14*n - 19)*x^2 + (36*n - 45)*x^3 + (44*n - 51)*x^4 + (26*n - 28)*x^5 + (6*n - 6)*x^6)*P[n - 1, x] + (x^2 + 7*x^3 + 18*x^4 + 22*x^5 + 13*x^6 + 3*x^7)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^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*n - 3)*x + (14*n - 19)*x^2 + (36*n - 45)*x^3 + (44*n - 51)*x^4 + (26*n - 28)*x^5 + (6*n - 6)*x^6)*v[i - 1] + (x^2 + 7*x^3 + 18*x^4 + 22*x^5 + 13*x^6 + 3*x^7)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^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

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