Real-rooted atlas

A267006 → (θ+1)²

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(7\,x^{4}\right)\,P_{n-2}(x) + \left(2\,x^{5}\right)\,P_{n-2}'(x) + \left(6\,x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
104913
2000162541
300000364985
400000006481145
5000000000100121221
600000000000144169313
70000000000000196225421

All 15 rows

Technical details
Node id
n-3013cc1bd15b292f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4694c9ea2eefc018

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, 4, 9;
  2: 0, 0, 0, 16, 25;
  3: 0, 0, 0,  0,  0, 36, 49;
  4: 0, 0, 0,  0,  0,  0,  0, 64, 81;
  5: 0, 0, 0,  0,  0,  0,  0,  0,  0, 100, 121;
  6: 0, 0, 0,  0,  0,  0,  0,  0,  0,   0,   0, 144, 169;
  7: 0, 0, 0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0, 196, 225;
  8: 0, 0, 0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0, 256, 289;
  9: 0, 0, 0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,   0,   0, 324, 361;
%F Formula
The row polynomials satisfy P(n,x) = 7*x^4*P(n-2,x) + 2*x^5*d/dx P(n-2,x) + 6*x^2*P(n-1,x) - 2*x^3*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x + 9*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[7*x^4*P[n - 2, x] + 2*x^5*D[P[n - 2, x], x] + 6*x^2*P[n - 1, x] - 2*x^3*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] = 1; v[2] = 4*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = 7*x^4*v[i - 2] + 2*x^5*deriv(v[i - 2], x) + 6*x^2*v[i - 1] - 2*x^3*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.