Real-rooted atlas

basis-map coefficients of A030528, shifted → basis A013610 → θ+1

only rational roots

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

Recurrence

\(\left(1 + 3\,n x\right)\,P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2} - 3\,n x - 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-1-6-7
21122740
3-1-18-81-108-208
41241624324051024
5-1-30-270-1080-2025-1458-4864
6136405216060758748510322528
7-1-42-567-3780-14175-30618-35721-17496-102400

All 15 rows

Technical details
Node id
n-e54041bba60d6610 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3151c955f16eaf46

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: -1,  -6;
  2:  1,  12,   27;
  3: -1, -18,  -81,  -108;
  4:  1,  24,  162,   432,    405;
  5: -1, -30, -270, -1080,  -2025,   -1458;
  6:  1,  36,  405,  2160,   6075,    8748,    5103;
  7: -1, -42, -567, -3780, -14175,  -30618,  -35721,  -17496;
  8:  1,  48,  756,  6048,  28350,   81648,  142884,  139968,   59049;
  9: -1, -54, -972, -9072, -51030, -183708, -428652, -629856, -531441, -196830;
%F Formula
The row polynomials satisfy (1+3*n*x)*P(n,x) = -(1+(3*n+6)*x+(9*n+9)*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[(-(1 + (3*n + 6)*x + (9*n + 9)*x^2)*P[n - 1, x])/(1 + 3*n*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] = (-(1 + (3*n + 6)*x + (9*n + 9)*x^2)*v[i - 1])/(1 + 3*n*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.