Real-rooted atlas

A136157 → (θ+1)²

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9\,x^{2}\right)\,P_{n-2}(x) + \left(4\,x^{3}\right)\,P_{n-2}'(x) + \left(4\,x\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 3\)
\(P_{1}(x) = 1 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
033
111213
2042731
30094857
4000167591
5000025108133
60000036147183
700000049192241
8000000064243307
90000000081300381
10000000000100363463
110000000000121432553
1200000000000144507651
13000000000000169588757
140000000000000196675871
Technical details
Node id
n-2f940f070dae5df4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-768c1cdd6f88e5b1

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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: 3;
  1: 1, 12;
  2: 0,  4, 27;
  3: 0,  0,  9, 48;
  4: 0,  0,  0, 16, 75;
  5: 0,  0,  0,  0, 25, 108;
  6: 0,  0,  0,  0,  0,  36, 147;
  7: 0,  0,  0,  0,  0,   0,  49, 192;
  8: 0,  0,  0,  0,  0,   0,   0,  64, 243;
  9: 0,  0,  0,  0,  0,   0,   0,   0,  81, 300;
%F Formula
The row polynomials satisfy P(n,x) = 9*x^2*P(n-2,x) + 4*x^3*d/dx P(n-2,x) + 4*x*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 1 + 12*x.
%t Mathematica
P[0, x_] := 3;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 2, x] + 4*x^3*D[P[n - 2, x], x] + 4*x*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = 9*x^2*v[i - 2] + 4*x^3*deriv(v[i - 2], x) + 4*x*v[i - 1] - 4*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 - 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.