Real-rooted atlas

A008292 → γ → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3Pn(1)
111
211
3145
411617
51444893
61104408513
71228216010883477
8148092161587225569

All 15 rows

Technical details
Node id
n-871a0925f3832687 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a78503eefaeb1d43

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
   3: 1,    4;
   4: 1,   16;
   5: 1,   44,     48;
   6: 1,  104,    408;
   7: 1,  228,   2160,   1088;
   8: 1,  480,   9216,  15872;
   9: 1,  988,  34848, 137216,  39680;
  10: 1, 2008, 122328, 920576, 884480;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^2-12*n)*x^2*P(n-2,x) - (2*n*x^2-8*n*x^3)*d/dx P(n-2,x) + (1+(4*n-8)*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n)*x^2*P[n - 2, x] - (2*n*x^2 - 8*n*x^3)*D[P[n - 2, x], x] + (1 + (4*n - 8)*x)*P[n - 1, x] + (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; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n)*x^2*v[i - 2] - (2*n*x^2 - 8*n*x^3)*deriv(v[i - 2], x) + (1 + (4*n - 8)*x)*v[i - 1] + (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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.