Real-rooted atlas

A133087 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,n x - n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 - 18\,x^{2} - 10\,x^{3} + 10\,n x + 17\,n x^{2} - 7\,n x^{3} - 3\,n x^{4} + 9\,n^{2} x^{2} + 15\,n^{2} x^{3} + 2\,n^{3} x^{3} + 3\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 15\,x^{2} - 16\,x^{3} - 5\,x^{4} - 6\,n x^{2} - 18\,n x^{3} - 15\,n x^{4} - 3\,n x^{5} - 2\,n^{2} x^{3} - 5\,n^{2} x^{4} - 3\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
21438
31815832
4181816548
511242644512176
6112458075367256
7116812002752169116896

All 15 rows

Technical details
Node id
n-2b3b998c7229ef50 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-34f3a95d22245511

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  4,   3;
  3: 1,  8,  15,   8;
  4: 1,  8,  18,  16,   5;
  5: 1, 12,  42,  64,  45,   12;
  6: 1, 12,  45,  80,  75,   36,   7;
  7: 1, 16,  81, 200, 275,  216,  91,  16;
  8: 1, 16,  84, 224, 350,  336, 196,  64,   9;
  9: 1, 20, 132, 448, 910, 1176, 980, 512, 153, 20;
%F Formula
The row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = (2+10*n*x+(9*n^2+17*n-18)*x^2+(2*n^3+15*n^2-7*n-10)*x^3+(3*n^3-3*n)*x^4)*P(n-1,x) - (4*x+(6*n+15)*x^2+(2*n^2+18*n+16)*x^3+(5*n^2+15*n+5)*x^4+(3*n^2+3*n)*x^5)*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[((2 + 10*n*x + (9*n^2 + 17*n - 18)*x^2 + (2*n^3 + 15*n^2 - 7*n - 10)*x^3 + (3*n^3 - 3*n)*x^4)*P[n - 1, x] - (4*x + (6*n + 15)*x^2 + (2*n^2 + 18*n + 16)*x^3 + (5*n^2 + 15*n + 5)*x^4 + (3*n^2 + 3*n)*x^5)*D[P[n - 1, x], x])/(2 + 2*n*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 10*n*x + (9*n^2 + 17*n - 18)*x^2 + (2*n^3 + 15*n^2 - 7*n - 10)*x^3 + (3*n^3 - 3*n)*x^4)*v[i - 1] - (4*x + (6*n + 15)*x^2 + (2*n^2 + 18*n + 16)*x^3 + (5*n^2 + 15*n + 5)*x^4 + (3*n^2 + 3*n)*x^5)*deriv(v[i - 1], x))/(2 + 2*n*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.