Real-rooted atlas

A201637 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(4 - 28\,x + 84\,x^{2} - 2\,n + 22\,n x - 94\,n x^{2} - 4\,n^{2} x + 34\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 14\,x^{2} + 12\,x^{3} - n x + 11\,n x^{2} - 10\,n x^{3} - 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-8 + 30\,x + 3\,n - 22\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 3\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 2\)

Rows ?

nx0x1x2x3x4x5Pn(1)
10
20
322
481220
52211672210
65265613324802520
711429041320014832360034650
8240112209636023256016992030240540540

All 15 rows

Technical details
Node id
n-0e0c25d6ebe9e558 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-69d5ef1a37e94893

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:    2;
   4:    8,     12;
   5:   22,    116,      72;
   6:   52,    656,    1332,      480;
   7:  114,   2904,   13200,    14832,     3600;
   8:  240,  11220,   96360,   232560,   169920,    30240;
   9:  494,  39900,  587400,  2576080,  3926520,  2046816,   282240;
  10: 1004, 134520, 3187500, 23062000, 62200320, 66157776, 26137440, 2903040;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(2*n-4+(4*n^2-22*n+28)*x+(4*n^3-34*n^2+94*n-84)*x^2)*P(n-2,x) - ((n-2)*x+(2*n^2-11*n+14)*x^2-(2*n^2-10*n+12)*x^3)*d/dx P(n-2,x) + (3*n-8+(4*n^2-22*n+30)*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) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (4*n^2 - 22*n + 28)*x + (4*n^3 - 34*n^2 + 94*n - 84)*x^2)*P[n - 2, x] - ((n - 2)*x + (2*n^2 - 11*n + 14)*x^2 - (2*n^2 - 10*n + 12)*x^3)*D[P[n - 2, x], x] + (3*n - 8 + (4*n^2 - 22*n + 30)*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] = 2; for(n = 4, N, my(i = n); v[i] = (-(2*n - 4 + (4*n^2 - 22*n + 28)*x + (4*n^3 - 34*n^2 + 94*n - 84)*x^2)*v[i - 2] - ((n - 2)*x + (2*n^2 - 11*n + 14)*x^2 - (2*n^2 - 10*n + 12)*x^3)*deriv(v[i - 2], x) + (3*n - 8 + (4*n^2 - 22*n + 30)*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. [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.