Real-rooted atlas

A274496 → D → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + 20\,x - 8\,x^{2} + 4\,n - 18\,n x + 4\,n x^{2} - 5\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-64 + 160\,x - 64\,x^{2} + 128\,n - 166\,n x + 54\,n x^{2} - 12\,n x^{3} - 75\,n^{2} + 59\,n^{2} x - 22\,n^{2} x^{2} + 4\,n^{2} x^{3} + 16\,n^{3} - 10\,n^{3} x + 4\,n^{3} x^{2} - n^{4} + n^{4} x\right)\,P_{n-1}(x) + \left(-16 + 32\,x + 16\,x^{2} - 32\,x^{3} + 32\,n - 20\,n x - 40\,n x^{2} + 4\,n x^{3} - 8\,n x^{4} - 18\,n^{2} - 8\,n^{2} x + 2\,n^{2} x^{2} - 8\,n^{2} x^{3} + 2\,n^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
222
344
416824
5321648
6729624192
714419248384
8256576384641280
951211527681282560
108002560288012801607680
11160051205760256032015360
12230496001536011520384038443008
134608192003072023040768076886016
1462723225667200716804032010752896229376
Technical details
Node id
n-1e8b302e37f8b9ac · revision 0 · JSON
Definition source
derived:one_plus_derivative
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:  16,    8;
  5:  32,   16;
  6:  72,   96,  24;
  7: 144,  192,  48;
  8: 256,  576, 384,  64;
  9: 512, 1152, 768, 128;
%F Formula
The row polynomials satisfy (n^3-5*n^2+4*n+4+(4*n^2-18*n+20)*x+(4*n-8)*x^2)*P(n,x) = -(n^4-16*n^3+75*n^2-128*n+64-(n^4-10*n^3+59*n^2-166*n+160)*x-(4*n^3-22*n^2+54*n-64)*x^2-(4*n^2-12*n)*x^3)*P(n-1,x) + (2*n^3-18*n^2+32*n-16-(8*n^2+20*n-32)*x-(2*n^3-2*n^2+40*n-16)*x^2-(8*n^2-4*n+32)*x^3-8*n*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[(-(n^4 - 16*n^3 + 75*n^2 - 128*n + 64 - (n^4 - 10*n^3 + 59*n^2 - 166*n + 160)*x - (4*n^3 - 22*n^2 + 54*n - 64)*x^2 - (4*n^2 - 12*n)*x^3)*P[n - 1, x] + (2*n^3 - 18*n^2 + 32*n - 16 - (8*n^2 + 20*n - 32)*x - (2*n^3 - 2*n^2 + 40*n - 16)*x^2 - (8*n^2 - 4*n + 32)*x^3 - 8*n*x^4)*D[P[n - 1, x], x])/(n^3 - 5*n^2 + 4*n + 4 + (4*n^2 - 18*n + 20)*x + (4*n - 8)*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] = (-(n^4 - 16*n^3 + 75*n^2 - 128*n + 64 - (n^4 - 10*n^3 + 59*n^2 - 166*n + 160)*x - (4*n^3 - 22*n^2 + 54*n - 64)*x^2 - (4*n^2 - 12*n)*x^3)*v[i - 1] + (2*n^3 - 18*n^2 + 32*n - 16 - (8*n^2 + 20*n - 32)*x - (2*n^3 - 2*n^2 + 40*n - 16)*x^2 - (8*n^2 - 4*n + 32)*x^3 - 8*n*x^4)*deriv(v[i - 1], x))/(n^3 - 5*n^2 + 4*n + 4 + (4*n^2 - 18*n + 20)*x + (4*n - 8)*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.