Real-rooted atlas

A089231 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x + x^{2} - n + 2\,n x + n^{2}\right)\,P_{n}(x) = \left(1 - 2\,x + x^{2} + n + 2\,n x - 2\,n x^{2} + 2\,n x^{3} + n^{2} + 4\,n^{2} x^{2} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-x^{2} - x^{4} - n x^{2} - 2\,n x^{3} - n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2718631
3138410824229
4212608407201201961
53163039008400540072019081
6431302132305460088200453605040207775
757240836456246960764400987840423360403202501801

All 15 rows

Technical details
Node id
n-b87d21ddf4200c60 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-333a237dceb0e170

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  3,    2;
  2:  7,   18,      6;
  3: 13,   84,    108,      24;
  4: 21,  260,    840,     720,      120;
  5: 31,  630,   3900,    8400,     5400,      720;
  6: 43, 1302,  13230,   54600,    88200,    45360,      5040;
  7: 57, 2408,  36456,  246960,   764400,   987840,    423360,     40320;
  8: 73, 4104,  86688,  874944,  4445280, 11007360,  11854080,   4354560,   362880;
  9: 91, 6570, 184680, 2600640, 19686240, 80015040, 165110400, 152409600, 48988800, 3628800;
%F Formula
The row polynomials satisfy (n^2-n+1+(2*n-2)*x+x^2)*P(n,x) = (n^2+n+1+(2*n^3+2*n-2)*x+(4*n^2-2*n+1)*x^2+2*n*x^3)*P(n-1,x) - ((n^2+n+1)*x^2+2*n*x^3+x^4)*d/dx 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[Together[((n^2 + n + 1 + (2*n^3 + 2*n - 2)*x + (4*n^2 - 2*n + 1)*x^2 + 2*n*x^3)*P[n - 1, x] - ((n^2 + n + 1)*x^2 + 2*n*x^3 + x^4)*D[P[n - 1, x], x])/(n^2 - n + 1 + (2*n - 2)*x + x^2)]];
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] = ((n^2 + n + 1 + (2*n^3 + 2*n - 2)*x + (4*n^2 - 2*n + 1)*x^2 + 2*n*x^3)*v[i - 1] - ((n^2 + n + 1)*x^2 + 2*n*x^3 + x^4)*deriv(v[i - 1], x))/(n^2 - n + 1 + (2*n - 2)*x + x^2)); 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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.