Real-rooted atlas

Veronese parent of A109446 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 2\,x + 2\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2} - 2\,n + n x + 11\,n x^{2} + 10\,n x^{3} + 2\,n x^{4} - 3\,n^{2} x + 6\,n^{2} x^{3} + 3\,n^{2} x^{4} - n^{3} x^{2} + n^{3} x^{4}\right)\,P_{n-1}(x) + \left(1 + 2\,x - 3\,x^{2} - 10\,x^{3} - 8\,x^{4} - 2\,x^{5} + 2\,n x + 3\,n x^{2} - 3\,n x^{3} - 7\,n x^{4} - 3\,n x^{5} + n^{2} x^{2} + n^{2} x^{3} - n^{2} x^{4} - n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
0022
1123
2026412
3169420
402122420664
51103040256112
602186010090428320
711463140175126498576

All 15 rows

Technical details
Node id
n-fa2b4efee20b7dcb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-da22bae9d59ddc25

Observed on the stored rows ?

  • Degrees: 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15 (irregular)
  • 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: 0,  2;
  1: 1,  2;
  2: 0,  2,   6,   4;
  3: 1,  6,   9,   4;
  4: 0,  2,  12,  24,  20,   6;
  5: 1, 10,  30,  40,  25,   6;
  6: 0,  2,  18,  60, 100,  90,  42,   8;
  7: 1, 14,  63, 140, 175, 126,  49,   8;
  8: 0,  2,  24, 112, 280, 420, 392, 224, 72, 10;
  9: 1, 18, 108, 336, 630, 756, 588, 288, 81, 10;
%F Formula
The row polynomials satisfy (2+(2*n+2)*x+(n^2+n)*x^2)*P(n,x) = -(2*n-2+(3*n^2-n-6)*x+(n^3-11*n-4)*x^2-(6*n^2+10*n)*x^3-(n^3+3*n^2+2*n)*x^4)*P(n-1,x) + (1+(2*n+2)*x+(n^2+3*n-3)*x^2+(n^2-3*n-10)*x^3-(n^2+7*n+8)*x^4-(n^2+3*n+2)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2*x.
%t Mathematica
P[0, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (3*n^2 - n - 6)*x + (n^3 - 11*n - 4)*x^2 - (6*n^2 + 10*n)*x^3 - (n^3 + 3*n^2 + 2*n)*x^4)*P[n - 1, x] + (1 + (2*n + 2)*x + (n^2 + 3*n - 3)*x^2 + (n^2 - 3*n - 10)*x^3 - (n^2 + 7*n + 8)*x^4 - (n^2 + 3*n + 2)*x^5)*D[P[n - 1, x], x])/(2 + (2*n + 2)*x + (n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2*x; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 2 + (3*n^2 - n - 6)*x + (n^3 - 11*n - 4)*x^2 - (6*n^2 + 10*n)*x^3 - (n^3 + 3*n^2 + 2*n)*x^4)*v[i - 1] + (1 + (2*n + 2)*x + (n^2 + 3*n - 3)*x^2 + (n^2 - 3*n - 10)*x^3 - (n^2 + 7*n + 8)*x^4 - (n^2 + 3*n + 2)*x^5)*deriv(v[i - 1], x))/(2 + (2*n + 2)*x + (n^2 + n)*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

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.