Real-rooted atlas

A274496 → D → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-16 + 4\,x - 4\,x^{2} + 8\,n - 10\,n x + 8\,n x^{2} + 4\,n^{2} x - 5\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-128 + 128\,x - 32\,x^{2} + 72\,n - 210\,n x + 78\,n x^{2} - 8\,n^{2} + 76\,n^{2} x - 73\,n^{2} x^{2} + 3\,n^{2} x^{3} - 6\,n^{3} x + 20\,n^{3} x^{2} - 4\,n^{3} x^{3} - n^{4} x^{2} + n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-32 + 16\,x + 32\,x^{2} - 16\,x^{3} + 8\,n - 52\,n x - 24\,n x^{2} + 36\,n x^{3} + 8\,n^{2} x - 18\,n^{2} x^{2} - 24\,n^{2} x^{3} + 2\,n^{2} x^{4} + 2\,n^{3} x^{2} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2Pn(1)
00
10
222
344
481624
5163248
6249672192
748192144384

All 15 rows

Technical details
Node id
n-3ea52383ec2ca4b4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b1032d52426bb000

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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;
  1:   0;
  2:   2;
  3:   4;
  4:   8,  16;
  5:  16,  32;
  6:  24,  96,   72;
  7:  48, 192,  144;
  8:  64, 384,  576, 256;
  9: 128, 768, 1152, 512;
%F Formula
The row polynomials satisfy (8*n-16+(4*n^2-10*n+4)*x+(n^3-5*n^2+8*n-4)*x^2)*P(n,x) = -(8*n^2-72*n+128+(6*n^3-76*n^2+210*n-128)*x+(n^4-20*n^3+73*n^2-78*n+32)*x^2-(n^4-4*n^3+3*n^2)*x^3)*P(n-1,x) + (8*n-32+(8*n^2-52*n+16)*x+(2*n^3-18*n^2-24*n+32)*x^2-(24*n^2-36*n+16)*x^3-(2*n^3-2*n^2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 72*n + 128 + (6*n^3 - 76*n^2 + 210*n - 128)*x + (n^4 - 20*n^3 + 73*n^2 - 78*n + 32)*x^2 - (n^4 - 4*n^3 + 3*n^2)*x^3)*P[n - 1, x] + (8*n - 32 + (8*n^2 - 52*n + 16)*x + (2*n^3 - 18*n^2 - 24*n + 32)*x^2 - (24*n^2 - 36*n + 16)*x^3 - (2*n^3 - 2*n^2)*x^4)*D[P[n - 1, x], x])/(8*n - 16 + (4*n^2 - 10*n + 4)*x + (n^3 - 5*n^2 + 8*n - 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] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(8*n^2 - 72*n + 128 + (6*n^3 - 76*n^2 + 210*n - 128)*x + (n^4 - 20*n^3 + 73*n^2 - 78*n + 32)*x^2 - (n^4 - 4*n^3 + 3*n^2)*x^3)*v[i - 1] + (8*n - 32 + (8*n^2 - 52*n + 16)*x + (2*n^3 - 18*n^2 - 24*n + 32)*x^2 - (24*n^2 - 36*n + 16)*x^3 - (2*n^3 - 2*n^2)*x^4)*deriv(v[i - 1], x))/(8*n - 16 + (4*n^2 - 10*n + 4)*x + (n^3 - 5*n^2 + 8*n - 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.