Real-rooted atlas

A236076 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(2 + 11\,x + 23\,x^{2} + 23\,x^{3} + 5\,x^{4}\right)\,P_{n-3}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 5\,x^{4}\right)\,P_{n-3}'(x) + \left(2 + 9\,x + 10\,x^{2} + 5\,x^{3}\right)\,P_{n-2}(x) + \left(x + 4\,x^{2} + 5\,x^{3}\right)\,P_{n-2}'(x) + \left(1 + 4\,x - 5\,x^{2}\right)\,P_{n-1}(x) + \left(-1 - 4\,x - 5\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
\(P_{2}(x) = 1 + 7\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
217311
30618529
4022239871
50012648013169
60034916215621391
70002016037929634889

All 15 rows

Technical details
Node id
n-7bc2010a5438fc74 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a5629228e3fedec0

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: 1;
  1: 2, 2;
  2: 1, 7,  3;
  3: 0, 6, 18,  5;
  4: 0, 2, 22, 39,   8;
  5: 0, 0, 12, 64,  80,  13;
  6: 0, 0,  3, 49, 162, 156,   21;
  7: 0, 0,  0, 20, 160, 379,  296,   34;
  8: 0, 0,  0,  4,  91, 456,  836,  549,   55;
  9: 0, 0,  0,  0,  30, 330, 1188, 1770, 1002, 89;
%F Formula
The row polynomials satisfy 2*P(n,x) = (2+11*x+23*x^2+23*x^3+5*x^4)*P(n-3,x) + (x+5*x^2+9*x^3+5*x^4)*d/dx P(n-3,x) + (2+9*x+10*x^2+5*x^3)*P(n-2,x) + (x+4*x^2+5*x^3)*d/dx P(n-2,x) + (1+4*x-5*x^2)*P(n-1,x) - (1+4*x+5*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 1 + 7*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 1 + 7*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + 11*x + 23*x^2 + 23*x^3 + 5*x^4)*P[n - 3, x] + (x + 5*x^2 + 9*x^3 + 5*x^4)*D[P[n - 3, x], x] + (2 + 9*x + 10*x^2 + 5*x^3)*P[n - 2, x] + (x + 4*x^2 + 5*x^3)*D[P[n - 2, x], x] + (1 + 4*x - 5*x^2)*P[n - 1, x] - (1 + 4*x + 5*x^2)*D[P[n - 1, x], x])/(2)]];
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] = 2 + 2*x; v[3] = 1 + 7*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2 + 11*x + 23*x^2 + 23*x^3 + 5*x^4)*v[i - 3] + (x + 5*x^2 + 9*x^3 + 5*x^4)*deriv(v[i - 3], x) + (2 + 9*x + 10*x^2 + 5*x^3)*v[i - 2] + (x + 4*x^2 + 5*x^3)*deriv(v[i - 2], x) + (1 + 4*x - 5*x^2)*v[i - 1] - (1 + 4*x + 5*x^2)*deriv(v[i - 1], x))/(2)); 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.