Real-rooted atlas

A274496 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + 6\,x + 4\,x^{2} + 5\,n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(-8 - 2\,x^{2} - 4\,x^{3} + 10\,n + 15\,n x + 6\,n x^{2} + 4\,n x^{3} + 6\,n^{2} + 4\,n^{2} x + 4\,n^{2} x^{2} + n^{3} x\right)\,P_{n-1}(x) + \left(-48 - 104\,x - 96\,x^{2} - 44\,x^{3} - 8\,x^{4} - 16\,n - 36\,n x - 30\,n x^{2} - 8\,n x^{3} - 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
2628
312416
43224460
564488120
6160192728432
732038414416864

All 15 rows

Technical details
Node id
n-bace070c6b73a0b3 · 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:    2;
  2:    6,    2;
  3:   12,    4;
  4:   32,   24,    4;
  5:   64,   48,    8;
  6:  160,  192,   72,   8;
  7:  320,  384,  144,  16;
  8:  768, 1280,  768, 192, 16;
  9: 1536, 2560, 1536, 384, 32;
%F Formula
The row polynomials satisfy (n^2+5*n-2+(4*n+6)*x+4*x^2)*P(n,x) = (6*n^2+10*n-8+(n^3+4*n^2+15*n)*x+(4*n^2+6*n-2)*x^2+(4*n-4)*x^3)*P(n-1,x) - (16*n+48+(4*n^2+36*n+104)*x+(2*n^2+30*n+96)*x^2+(8*n+44)*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[((6*n^2 + 10*n - 8 + (n^3 + 4*n^2 + 15*n)*x + (4*n^2 + 6*n - 2)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] - (16*n + 48 + (4*n^2 + 36*n + 104)*x + (2*n^2 + 30*n + 96)*x^2 + (8*n + 44)*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] = ((6*n^2 + 10*n - 8 + (n^3 + 4*n^2 + 15*n)*x + (4*n^2 + 6*n - 2)*x^2 + (4*n - 4)*x^3)*v[i - 1] - (16*n + 48 + (4*n^2 + 36*n + 104)*x + (2*n^2 + 30*n + 96)*x^2 + (8*n + 44)*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.