Real-rooted atlas

A152198 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3\right)\,P_{n}(x) = \left(-9 - 9\,x + 12\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-40 - 54\,x - 18\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2538
3538
42530964
52530964
612522513527512
712522513527512
862515001350540814096
962515001350540814096
1031259375112506750202524332768
1131259375112506750202524332768
1215625562508437567500303757290729262144
1315625562508437567500303757290729262144
14781253281255906255906253543751275752551521872097152

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(5 + 3\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-86bc5e5caf1f46e7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-28b2abc69ae4dafc)
Fingerprint
f6-aee9d583da3df3c2

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, 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:   1;
  2:   5,    3;
  3:   5,    3;
  4:  25,   30,    9;
  5:  25,   30,    9;
  6: 125,  225,  135,  27;
  7: 125,  225,  135,  27;
  8: 625, 1500, 1350, 540, 81;
  9: 625, 1500, 1350, 540, 81;
%F Formula
The row polynomials satisfy 3*P(n,x) = (12*n-9+(9*n-9)*x)*P(n-1,x) - (40+54*x+18*x^2)*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[((12*n - 9 + (9*n - 9)*x)*P[n - 1, x] - (40 + 54*x + 18*x^2)*D[P[n - 1, x], 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] = ((12*n - 9 + (9*n - 9)*x)*v[i - 1] - (40 + 54*x + 18*x^2)*deriv(v[i - 1], 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.