Real-rooted atlas

A163936 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,x + n\right)\,P_{n}(x) = \left(14\,x - 18\,x^{3} + 4\,x^{4} - 2\,n - 23\,n x + n x^{2} + 14\,n x^{3} - 2\,n x^{4} + 3\,n^{2} + 11\,n^{2} x + 3\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 2\,x^{2} + 12\,x^{3} - 10\,x^{4} + 2\,x^{5} + 3\,n x - 6\,n x^{3} + 3\,n x^{4} - n^{2} x + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 + 15\,x - 14\,x^{2} + 2\,x^{3} - 5\,n - 7\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 5\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2314
31410125
482102251210
556411004845612205
644281250887561908119127720
7390241502641545005456068102461405405

All 15 rows

Technical details
Node id
n-8e3779e7f0bd1f37 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-853f0e1aa93cb97b

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:       1;
  2:       3,        1;
  3:      14,       10,        1;
  4:      82,      102,       25,        1;
  5:     564,     1100,      484,       56,        1;
  6:    4428,    12508,     8756,     1908,      119,       1;
  7:   39024,   150264,   154500,    54560,     6810,     246,     1;
  8:  381456,  1911744,  2717364,  1427220,   295550,   22914,   501,    1;
  9: 4096800, 25786512, 48346488, 35502064, 11078000, 1466060, 74288, 1012, 1;
%F Formula
The row polynomials satisfy (n-3+2*x)*P(n,x) = -(n^3-3*n^2+2*n+(2*n^3-11*n^2+23*n-14)*x+(n^3-3*n^2-n)*x^2+(3*n^2-14*n+18)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((n^2-3*n+2)*x+2*x^2-(n^2-6*n+12)*x^3-(3*n-10)*x^4-2*x^5)*d/dx P(n-2,x) + (2*n^2-5*n-1+(2*n^2-7*n+15)*x+(5*n-14)*x^2+2*x^3)*P(n-1,x) + ((n-3)*x-(n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 + 2*n + (2*n^3 - 11*n^2 + 23*n - 14)*x + (n^3 - 3*n^2 - n)*x^2 + (3*n^2 - 14*n + 18)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((n^2 - 3*n + 2)*x + 2*x^2 - (n^2 - 6*n + 12)*x^3 - (3*n - 10)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - 5*n - 1 + (2*n^2 - 7*n + 15)*x + (5*n - 14)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 3)*x - (n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 3 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 3*n^2 + 2*n + (2*n^3 - 11*n^2 + 23*n - 14)*x + (n^3 - 3*n^2 - n)*x^2 + (3*n^2 - 14*n + 18)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((n^2 - 3*n + 2)*x + 2*x^2 - (n^2 - 6*n + 12)*x^3 - (3*n - 10)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - 5*n - 1 + (2*n^2 - 7*n + 15)*x + (5*n - 14)*x^2 + 2*x^3)*v[i - 1] + ((n - 3)*x - (n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 3 + 2*x)); 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

Same fingerprint ?

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.