Real-rooted atlas

A330510 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(9 + 3\,x - 9\,x^{2}\right)\,P_{n-3}(x) + \left(1 - 3\,x - 3\,x^{2} - 4\,x^{3}\right)\,P_{n-3}'(x) + \left(-22 - 2\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(-2 + 6\,x + 8\,x^{2}\right)\,P_{n-2}'(x) + \left(15 + 8\,x\right)\,P_{n-1}(x) + \left(1 - 4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 6 + 3\,x\)
\(P_{2}(x) = 14 + 18\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1639
21418436
32766375135
446180190646486
57240768443110071701
61068101958195884614685832
714914704784686546981505203919683

All 15 rows

Technical details
Node id
n-4d2ad51b1249d571 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-cf42a42d64689032

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:   2;
  1:   6,    3;
  2:  14,   18,     4;
  3:  27,   66,    37,     5;
  4:  46,  180,   190,    64,     6;
  5:  72,  407,   684,   431,   100,     7;
  6: 106,  810,  1958,  1958,   846,   146,     8;
  7: 149, 1470,  4784,  6865,  4698,  1505,   203,    9;
  8: 202, 2488, 10412, 20072, 19686,  9976,  2492,  272,  10;
  9: 266, 3987, 20748, 51322, 67542, 49021, 19356, 3906, 354, 11;
%F Formula
The row polynomials satisfy 4*P(n,x) = (9+3*x-9*x^2)*P(n-3,x) + (1-3*x-3*x^2-4*x^3)*d/dx P(n-3,x) - (22+2*x+4*x^2)*P(n-2,x) - (2-6*x-8*x^2)*d/dx P(n-2,x) + (15+8*x)*P(n-1,x) + (1-4*x)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 6 + 3*x, P(2,x) = 14 + 18*x + 4*x^2.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 6 + 3*x;
P[2, x_] := 14 + 18*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9 + 3*x - 9*x^2)*P[n - 3, x] + (1 - 3*x - 3*x^2 - 4*x^3)*D[P[n - 3, x], x] - (22 + 2*x + 4*x^2)*P[n - 2, x] - (2 - 6*x - 8*x^2)*D[P[n - 2, x], x] + (15 + 8*x)*P[n - 1, x] + (1 - 4*x)*D[P[n - 1, x], x])/(4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 6 + 3*x; v[3] = 14 + 18*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((9 + 3*x - 9*x^2)*v[i - 3] + (1 - 3*x - 3*x^2 - 4*x^3)*deriv(v[i - 3], x) - (22 + 2*x + 4*x^2)*v[i - 2] - (2 - 6*x - 8*x^2)*deriv(v[i - 2], x) + (15 + 8*x)*v[i - 1] + (1 - 4*x)*deriv(v[i - 1], x))/(4)); 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. [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.