Real-rooted atlas

A133080 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 4\,n x - n x^{2} + 5\,n^{2} - 6\,n^{2} x + n^{2} x^{2} - 4\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(6\,x^{2} - 2\,x^{3} - 6\,n + 14\,n x - 9\,n x^{2} + n x^{3} + 17\,n^{2} - 19\,n^{2} x - 4\,n^{2} x^{2} + 3\,n^{2} x^{3} - 17\,n^{3} + 11\,n^{3} x^{2} - 2\,n^{3} x^{3} + 7\,n^{4} + 7\,n^{4} x - 4\,n^{4} x^{2} - n^{5} - 2\,n^{5} x\right)\,P_{n-1}(x) + \left(2\,x^{3} - x^{4} - 2\,n x + 5\,n x^{2} - 3\,n x^{3} + 5\,n^{2} x - 5\,n^{2} x^{2} - 3\,n^{2} x^{3} + 2\,n^{2} x^{4} - 4\,n^{3} x - 2\,n^{3} x^{2} + 4\,n^{3} x^{3} + n^{4} x + 2\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
20639
30615425
400020525
50002035661
60000042749
70000042638113

All 15 rows

Technical details
Node id
n-9f0cb6e392f8c3f1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0ea1e646a8483d38

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: 3, 2;
  2: 0, 6,  3;
  3: 0, 6, 15,  4;
  4: 0, 0,  0, 20,  5;
  5: 0, 0,  0, 20, 35,  6;
  6: 0, 0,  0,  0,  0, 42,  7;
  7: 0, 0,  0,  0,  0, 42, 63,  8;
  8: 0, 0,  0,  0,  0,  0,  0, 72,  9;
  9: 0, 0,  0,  0,  0,  0,  0, 72, 99, 10;
%F Formula
The row polynomials satisfy (n^4-4*n^3+5*n^2-2*n+(2*n^3-6*n^2+4*n)*x+(n^2-n)*x^2)*P(n,x) = -(n^5-7*n^4+17*n^3-17*n^2+6*n+(2*n^5-7*n^4+19*n^2-14*n)*x+(4*n^4-11*n^3+4*n^2+9*n-6)*x^2+(2*n^3-3*n^2-n+2)*x^3)*P(n-1,x) + ((n^4-4*n^3+5*n^2-2*n)*x+(2*n^4-2*n^3-5*n^2+5*n)*x^2+(4*n^3-3*n^2-3*n+2)*x^3+(2*n^2-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 7*n^4 + 17*n^3 - 17*n^2 + 6*n + (2*n^5 - 7*n^4 + 19*n^2 - 14*n)*x + (4*n^4 - 11*n^3 + 4*n^2 + 9*n - 6)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*P[n - 1, x] + ((n^4 - 4*n^3 + 5*n^2 - 2*n)*x + (2*n^4 - 2*n^3 - 5*n^2 + 5*n)*x^2 + (4*n^3 - 3*n^2 - 3*n + 2)*x^3 + (2*n^2 - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^3 + 5*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*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; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - 7*n^4 + 17*n^3 - 17*n^2 + 6*n + (2*n^5 - 7*n^4 + 19*n^2 - 14*n)*x + (4*n^4 - 11*n^3 + 4*n^2 + 9*n - 6)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*v[i - 1] + ((n^4 - 4*n^3 + 5*n^2 - 2*n)*x + (2*n^4 - 2*n^3 - 5*n^2 + 5*n)*x^2 + (4*n^3 - 3*n^2 - 3*n + 2)*x^3 + (2*n^2 - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^3 + 5*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*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.