Real-rooted atlas

basis-map coefficients of A113214, shifted → x ↦ x+1 → D+1

only rational roots

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

Recurrence

\(\left(1 + x + n\right)\,P_{n}(x) = \left(-4 - 4\,x - x^{2} - 2\,n - n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-3-1-4
286115
3-20-24-9-1-54
4488048121189
5-112-240-200-80-15-1-648
62566727204001201812187
7-576-1792-2352-1680-700-168-21-1-7290

All 15 rows

Technical details
Node id
n-de5e14d043ee5c31 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • 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:    -3,     -1;
  2:     8,      6,      1;
  3:   -20,    -24,     -9,     -1;
  4:    48,     80,     48,     12,      1;
  5:  -112,   -240,   -200,    -80,    -15,    -1;
  6:   256,    672,    720,    400,    120,    18,     1;
  7:  -576,  -1792,  -2352,  -1680,   -700,  -168,   -21,   -1;
  8:  1280,   4608,   7168,   6272,   3360,  1120,   224,   24,   1;
  9: -2816, -11520, -20736, -21504, -14112, -6048, -1680, -288, -27, -1;
%F Formula
The row polynomials satisfy (n+1+x)*P(n,x) = -(2*n+4+(n+4)*x+x^2)*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[(-(2*n + 4 + (n + 4)*x + x^2)*P[n - 1, x])/(n + 1 + x)]];
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] = (-(2*n + 4 + (n + 4)*x + x^2)*v[i - 1])/(n + 1 + x)); 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.