Real-rooted atlas

A322944 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 + 8\,x + 9\,x^{2} + 7\,n - 4\,n x - n^{2}\right)\,P_{n-2}(x) + \left(12\,x - 8\,x^{2} + 4\,x^{3} - 6\,n x + 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(-9\,x^{2} + 12\,x^{3}\right)\,P_{n-2}''(x) + \left(-6 + 4\,x + 2\,n\right)\,P_{n-1}(x) + \left(6\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2145
3224935
4615213516309
52410881692448253277
6120880021060928011253640421
77207968027252017664034750237649566735
85040799680372204033107209362501023124459648880565

All 15 rows

Technical details
Node id
n-be8e1f4bb575779f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7d7c17087b843f8b

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:      1,         4;
   3:      2,        24,         9;
   4:      6,       152,       135,         16;
   5:     24,      1088,      1692,        448,        25;
   6:    120,      8800,     21060,       9280,      1125,        36;
   7:    720,     79680,    272520,     176640,     34750,      2376,       49;
   8:   5040,    799680,   3722040,    3310720,    936250,    102312,     4459,     64;
   9:  40320,   8816640,  53907840,   62899200,  23898000,   3701376,   255192,   7680,    81;
  10: 362880, 105960960, 828455040, 1228021760, 600138000, 123246144, 11928168, 563712, 12393, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+10+(4*n-8)*x-9*x^2)*P(n-2,x) - ((6*n-12)*x-(4*n-8)*x^2-4*x^3)*d/dx P(n-2,x) - (9*x^2-12*x^3)*d^2/dx^2 P(n-2,x) + (2*n-6+4*x)*P(n-1,x) + (6*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 10 + (4*n - 8)*x - 9*x^2)*P[n - 2, x] - ((6*n - 12)*x - (4*n - 8)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (9*x^2 - 12*x^3)*D[P[n - 2, x], {x, 2}] + (2*n - 6 + 4*x)*P[n - 1, x] + (6*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] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 7*n + 10 + (4*n - 8)*x - 9*x^2)*v[i - 2] - ((6*n - 12)*x - (4*n - 8)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (9*x^2 - 12*x^3)*deriv(deriv(v[i - 2], x), x) + (2*n - 6 + 4*x)*v[i - 1] + (6*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.