Real-rooted atlas

A128540 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(14\,x^{3}\right)\,P_{n-3}(x) + \left(5\,x^{4}\right)\,P_{n-3}'(x) + \left(6\,x^{2}\right)\,P_{n-2}(x) + \left(5\,x^{3}\right)\,P_{n-2}'(x) + \left(8\,x\right)\,P_{n-1}(x) + \left(-5\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 4\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
204610
30091221
4000202545
50000404888
6000007891169
7000000147168315

All 15 rows

Technical details
Node id
n-56c785d2b08d9301 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6367169308476412

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: 1;
  1: 1, 2;
  2: 0, 4, 6;
  3: 0, 0, 9, 12;
  4: 0, 0, 0, 20, 25;
  5: 0, 0, 0,  0, 40, 48;
  6: 0, 0, 0,  0,  0, 78,  91;
  7: 0, 0, 0,  0,  0,  0, 147, 168;
  8: 0, 0, 0,  0,  0,  0,   0, 272, 306;
  9: 0, 0, 0,  0,  0,  0,   0,   0, 495, 550;
%F Formula
The row polynomials satisfy 2*P(n,x) = 14*x^3*P(n-3,x) + 5*x^4*d/dx P(n-3,x) + 6*x^2*P(n-2,x) + 5*x^3*d/dx P(n-2,x) + 8*x*P(n-1,x) - 5*x^2*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 4*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 4*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(14*x^3*P[n - 3, x] + 5*x^4*D[P[n - 3, x], x] + 6*x^2*P[n - 2, x] + 5*x^3*D[P[n - 2, x], x] + 8*x*P[n - 1, x] - 5*x^2*D[P[n - 1, x], x])/(2)]];
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] = 1 + 2*x; v[3] = 4*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (14*x^3*v[i - 3] + 5*x^4*deriv(v[i - 3], x) + 6*x^2*v[i - 2] + 5*x^3*deriv(v[i - 2], x) + 8*x*v[i - 1] - 5*x^2*deriv(v[i - 1], x))/(2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.