Real-rooted atlas

A152198 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 6\,x + 4\,x^{2} + 5\,n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(-8 - 12\,x - 10\,x^{2} - 4\,x^{3} + 4\,n + 13\,n x + 10\,n x^{2} + 4\,n x^{3} + 7\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-72 - 156\,x - 132\,x^{2} - 52\,x^{3} - 8\,x^{4} - 36\,n - 62\,n x - 38\,n x^{2} - 8\,n x^{3} - 4\,n^{2} - 6\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2314
3314
486115
586115
620249154
720249154

All 15 rows

Technical details
Node id
n-0c08737b4d735088 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8eac50fee3ed81bc

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:  3,  1;
  3:  3,  1;
  4:  8,  6,  1;
  5:  8,  6,  1;
  6: 20, 24,  9,  1;
  7: 20, 24,  9,  1;
  8: 48, 80, 48, 12, 1;
  9: 48, 80, 48, 12, 1;
%F Formula
The row polynomials satisfy (n^2+5*n-2+(4*n+6)*x+4*x^2)*P(n,x) = (n^3+7*n^2+4*n-8+(n^3+8*n^2+13*n-12)*x+(4*n^2+10*n-10)*x^2+(4*n-4)*x^3)*P(n-1,x) - (4*n^2+36*n+72+(6*n^2+62*n+156)*x+(2*n^2+38*n+132)*x^2+(8*n+52)*x^3+8*x^4)*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[((n^3 + 7*n^2 + 4*n - 8 + (n^3 + 8*n^2 + 13*n - 12)*x + (4*n^2 + 10*n - 10)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] - (4*n^2 + 36*n + 72 + (6*n^2 + 62*n + 156)*x + (2*n^2 + 38*n + 132)*x^2 + (8*n + 52)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(n^2 + 5*n - 2 + (4*n + 6)*x + 4*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] = ((n^3 + 7*n^2 + 4*n - 8 + (n^3 + 8*n^2 + 13*n - 12)*x + (4*n^2 + 10*n - 10)*x^2 + (4*n - 4)*x^3)*v[i - 1] - (4*n^2 + 36*n + 72 + (6*n^2 + 62*n + 156)*x + (2*n^2 + 38*n + 132)*x^2 + (8*n + 52)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(n^2 + 5*n - 2 + (4*n + 6)*x + 4*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. [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.