Real-rooted atlas

A269944 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21326
318171036
411897180100396
51354031829316017007128
6161131312307507148386044200192456
71983570608885060731960278314702016354007313328

All 15 rows

Technical details
Node id
n-39aebdc41ec576d0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e90640f2cfea9e20

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): 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,   1;
  2: 1,   3,     2;
  3: 1,   8,    17,     10;
  4: 1,  18,    97,    180,      100;
  5: 1,  35,   403,   1829,     3160,      1700;
  6: 1,  61,  1313,  12307,    50714,     83860,      44200;
  7: 1,  98,  3570,  60888,   506073,   1960278,    3147020,    1635400;
  8: 1, 148,  8470, 239388,  3550473,  27263928,  101160920,  158986400,    81770000;
  9: 1, 213, 18090, 789938, 19110693, 258044673, 1873316240, 6734446200, 10415886000, 5315050000;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n^2-2*n+2)*x)*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[(1 + (n^2 - 2*n + 2)*x)*P[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] = (1 + (n^2 - 2*n + 2)*x)*v[i - 1]); 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.