Real-rooted atlas

A085880 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\,n - n x + n^{2} x\right)\,P_{n}(x) = \left(-24 - 12\,x + 16\,n - 4\,n x - 6\,n x^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3816630
4401209020270
52248961008448702646
613446720100806720210025227216
78448506889504084480396009504924288684
8549123843848648649609606006002162164204234323127410

All 15 rows

Technical details
Node id
n-042d5cf9e98e7765 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e620d4a356abb37c

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       2,        2;
   3:       8,       16,        6;
   4:      40,      120,       90,        20;
   5:     224,      896,     1008,       448,       70;
   6:    1344,     6720,    10080,      6720,     2100,      252;
   7:    8448,    50688,    95040,     84480,    39600,     9504,      924;
   8:   54912,   384384,   864864,    960960,   600600,   216216,    42042,    3432;
   9:  366080,  2928640,  7687680,  10250240,  8008000,  3843840,  1121120,  183040,  12870;
  10: 2489344, 22404096, 67212288, 104552448, 98017920, 58810752, 22870848, 5601024, 787644, 48620;
%F Formula
The row polynomials satisfy (2*n+(n^2-n)*x)*P(n,x) = (16*n-24+(8*n^2-4*n-12)*x+(4*n^2-6*n)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 24 + (8*n^2 - 4*n - 12)*x + (4*n^2 - 6*n)*x^2)*P[n - 1, x])/(2*n + (n^2 - n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n - 24 + (8*n^2 - 4*n - 12)*x + (4*n^2 - 6*n)*x^2)*v[i - 1])/(2*n + (n^2 - n)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.