Real-rooted atlas

A152198 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(32 + 12\,x + 8\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(-56\,x - 24\,x^{2} - 2\,x^{3} + 32\,n + 64\,n x + 8\,n x^{2} - n x^{3} + 12\,n^{2} x + 17\,n^{2} x^{2} + 2\,n^{2} x^{3} + n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-64 - 176\,x - 136\,x^{2} - 40\,x^{3} - 4\,x^{4} - 32\,n x - 68\,n x^{2} - 38\,n x^{3} - 6\,n x^{4} - 4\,n^{2} x^{2} - 6\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2224
3224
448315
548315
682418454
782418454
8166472325189
9166472325189
1032160240160506648
1132160240160506648
12643847206403007272187
13643847206403007272187
141288962016224014005049887290
Technical details
Node id
n-a12b13f8ef19e579 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-90678fce5eba4992

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:  2,  2;
  3:  2,  2;
  4:  4,  8,  3;
  5:  4,  8,  3;
  6:  8, 24, 18,  4;
  7:  8, 24, 18,  4;
  8: 16, 64, 72, 32, 5;
  9: 16, 64, 72, 32, 5;
%F Formula
The row polynomials satisfy (32+(8*n+12)*x+(n^2+n)*x^2)*P(n,x) = (32*n+(12*n^2+64*n-56)*x+(n^3+17*n^2+8*n-24)*x^2+(n^3+2*n^2-n-2)*x^3)*P(n-1,x) - (64+(32*n+176)*x+(4*n^2+68*n+136)*x^2+(6*n^2+38*n+40)*x^3+(2*n^2+6*n+4)*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[((32*n + (12*n^2 + 64*n - 56)*x + (n^3 + 17*n^2 + 8*n - 24)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] - (64 + (32*n + 176)*x + (4*n^2 + 68*n + 136)*x^2 + (6*n^2 + 38*n + 40)*x^3 + (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(32 + (8*n + 12)*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; for(n = 1, N, my(i = n + 1); v[i] = ((32*n + (12*n^2 + 64*n - 56)*x + (n^3 + 17*n^2 + 8*n - 24)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] - (64 + (32*n + 176)*x + (4*n^2 + 68*n + 136)*x^2 + (6*n^2 + 38*n + 40)*x^3 + (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(32 + (8*n + 12)*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

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.