Real-rooted atlas

A173018 → x ↦ x+1 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(12 + 32\,x + 24\,x^{2} - 16\,n - 40\,n x - 26\,n x^{2} + 7\,n^{2} + 16\,n^{2} x + 9\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x + 10\,x^{2} + 6\,x^{3} - 4\,n x - 9\,n x^{2} - 5\,n x^{3} + n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(10 + 18\,x - 9\,n - 12\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x + 3\,x^{2} - n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1\)

Rows ?

nx0x1x2x3x4x5Pn(1)
10
20
311
4628
53628367
6240300904634
718003120162024856793
81512033600252007224630681780

All 15 rows

Technical details
Node id
n-566cddcc2128607a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-867bd83b37df747f

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
  • 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:       0;
   2:       0;
   3:       1;
   4:       6,       2;
   5:      36,      28,       3;
   6:     240,     300,      90,       4;
   7:    1800,    3120,    1620,     248,      5;
   8:   15120,   33600,   25200,    7224,    630,      6;
   9:  141120,  383040,  378000,  163296,  28980,   1524,    7;
  10: 1451520, 4656960, 5715360, 3336480, 932400, 108900, 3570, 8;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(n^3-7*n^2+16*n-12+(2*n^3-16*n^2+40*n-32)*x+(n^3-9*n^2+26*n-24)*x^2)*P(n-2,x) + ((n^2-4*n+4)*x+(2*n^2-9*n+10)*x^2+(n^2-5*n+6)*x^3)*d/dx P(n-2,x) + (2*n^2-9*n+10+(2*n^2-12*n+18)*x)*P(n-1,x) - ((n-3)*x+(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 7*n^2 + 16*n - 12 + (2*n^3 - 16*n^2 + 40*n - 32)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*P[n - 2, x] + ((n^2 - 4*n + 4)*x + (2*n^2 - 9*n + 10)*x^2 + (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 9*n + 10 + (2*n^2 - 12*n + 18)*x)*P[n - 1, x] - ((n - 3)*x + (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = (-(n^3 - 7*n^2 + 16*n - 12 + (2*n^3 - 16*n^2 + 40*n - 32)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*v[i - 2] + ((n^2 - 4*n + 4)*x + (2*n^2 - 9*n + 10)*x^2 + (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 9*n + 10 + (2*n^2 - 12*n + 18)*x)*v[i - 1] - ((n - 3)*x + (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); 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. [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.