Real-rooted atlas

A122076 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1235
228717
3215301865
42248010447257
52351703553401231025
6248315932141010683224097
7263532207943965208326284316385

All 15 rows

Technical details
Node id
n-84b60802c7175ea1 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f83cf33d2c000fbc

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: 2,  3;
  2: 2,  8,    7;
  3: 2, 15,   30,   18;
  4: 2, 24,   80,  104,    47;
  5: 2, 35,  170,  355,   340,   123;
  6: 2, 48,  315,  932,  1410,  1068,   322;
  7: 2, 63,  532, 2079,  4396,  5208,  3262,   843;
  8: 2, 80,  840, 4144, 11440, 18784, 18280,  9760,  2207;
  9: 2, 99, 1260, 7602, 26226, 55809, 74838, 61785, 28746, 5778;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (2*n-2-(n-1)*x-(5*n-5)*x^2)*P(n-1,x) + (4*x+9*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 2 + 3*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 - (n - 1)*x - (5*n - 5)*x^2)*P[n - 1, x] + (4*x + 9*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 - (n - 1)*x - (5*n - 5)*x^2)*v[i - 1] + (4*x + 9*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n - 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

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