Real-rooted atlas

A008277 → θ+1 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 + 10\,x - 20\,x^{2} - 2\,n x + 9\,n x^{2} - n^{2} x^{2}\right)\,P_{n-4}(x) + \left(x^{2} - 4\,x^{3} + n x^{3}\right)\,P_{n-4}'(x) + \left(4 - 22\,x + 12\,x^{2} + 6\,n x - 11\,n x^{2} + 2\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-3\,x^{2} + 4\,x^{3} - 2\,n x^{3}\right)\,P_{n-3}'(x) + \left(-6 + 14\,x - 6\,n x + 2\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(3\,x^{2} + n x^{3}\right)\,P_{n-2}'(x) + \left(4 - 2\,x + 2\,n x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 2\,x\)
\(P_{3}(x) = 3 + 12\,x + 3\,x^{2}\)
\(P_{4}(x) = 4 + 36\,x + 42\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3312318
443642486
55802251205435
66150780108031062332
77252210056004515756713237
8839247882100034020173881778879382

All 15 rows

Technical details
Node id
n-679fa25ac998751a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-40d9db5c06f10661

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:  2,   2;
   3:  3,  12,     3;
   4:  4,  36,    42,      4;
   5:  5,  80,   225,    120,      5;
   6:  6, 150,   780,   1080,    310,       6;
   7:  7, 252,  2100,   5600,   4515,     756,      7;
   8:  8, 392,  4788,  21000,  34020,   17388,   1778,      8;
   9:  9, 576,  9702,  63504, 173775,  186480,  63525,   4080,    9;
  10: 10, 810, 18000, 164640, 684810, 1275750, 954940, 223920, 9198, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(2*n-10)*x+(n^2-9*n+20)*x^2)*P(n-4,x) + (x^2+(n-4)*x^3)*d/dx P(n-4,x) + (4+(6*n-22)*x+(2*n^2-11*n+12)*x^2)*P(n-3,x) - (3*x^2+(2*n-4)*x^3)*d/dx P(n-3,x) - (6+(6*n-14)*x+(n^2-2*n)*x^2)*P(n-2,x) + (3*x^2+n*x^3)*d/dx P(n-2,x) + (4+(2*n-2)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 2*x, P(3,x) = 3 + 12*x + 3*x^2, P(4,x) = 4 + 36*x + 42*x^2 + 4*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[3, x_] := 3 + 12*x + 3*x^2;
P[4, x_] := 4 + 36*x + 42*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 10)*x + (n^2 - 9*n + 20)*x^2)*P[n - 4, x] + (x^2 + (n - 4)*x^3)*D[P[n - 4, x], x] + (4 + (6*n - 22)*x + (2*n^2 - 11*n + 12)*x^2)*P[n - 3, x] - (3*x^2 + (2*n - 4)*x^3)*D[P[n - 3, x], x] - (6 + (6*n - 14)*x + (n^2 - 2*n)*x^2)*P[n - 2, x] + (3*x^2 + n*x^3)*D[P[n - 2, x], x] + (4 + (2*n - 2)*x)*P[n - 1, x] - 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] = 2 + 2*x; v[3] = 3 + 12*x + 3*x^2; v[4] = 4 + 36*x + 42*x^2 + 4*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + (2*n - 10)*x + (n^2 - 9*n + 20)*x^2)*v[i - 4] + (x^2 + (n - 4)*x^3)*deriv(v[i - 4], x) + (4 + (6*n - 22)*x + (2*n^2 - 11*n + 12)*x^2)*v[i - 3] - (3*x^2 + (2*n - 4)*x^3)*deriv(v[i - 3], x) - (6 + (6*n - 14)*x + (n^2 - 2*n)*x^2)*v[i - 2] + (3*x^2 + n*x^3)*deriv(v[i - 2], x) + (4 + (2*n - 2)*x)*v[i - 1] - 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. [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.