Real-rooted atlas

A035469 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-30 + 20\,x + 9\,x^{2} + 33\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(10\,x - 20\,x^{2} + 4\,x^{3} - 6\,n x + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(-8 + 4\,x + 6\,n\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2448
32848985
4280640216161152
536401008046806402519065
65824018592010692020480150036373096
7110656039513602663640640640665003024498431773
8243443209530752072817920207513602730000177408548864216134080

All 15 rows

Technical details
Node id
n-9a51e72a74f2d3e0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b01a90078e4aa5ae

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:           1;
   2:           4,           4;
   3:          28,          48,           9;
   4:         280,         640,         216,          16;
   5:        3640,       10080,        4680,         640,         25;
   6:       58240,      185920,      106920,       20480,       1500,        36;
   7:     1106560,     3951360,     2663640,      640640,      66500,      3024,       49;
   8:    24344320,    95307520,    72817920,    20751360,    2730000,    177408,     5488,     64;
   9:   608608000,  2575372800,  2180525760,   710492160,  111594000,   9253440,   411600,   9216,    81;
  10: 17041024000, 77120243200, 71210361600, 25901747200, 4681152000, 466058880, 26589360, 860160, 14580, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-33*n+30+(12*n-20)*x-9*x^2)*P(n-2,x) - ((6*n-10)*x-(12*n-20)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (6*n-8+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 33*n + 30 + (12*n - 20)*x - 9*x^2)*P[n - 2, x] - ((6*n - 10)*x - (12*n - 20)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (6*n - 8 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 4*x; for(n = 3, N, my(i = n); v[i] = -(9*n^2 - 33*n + 30 + (12*n - 20)*x - 9*x^2)*v[i - 2] - ((6*n - 10)*x - (12*n - 20)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (6*n - 8 + 4*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], 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

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.