Real-rooted atlas

A163936 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + x + n\right)\,P_{n}(x) = \left(15\,x^{2} - 18\,x^{3} + 3\,x^{4} - 31\,n x^{2} + 21\,n x^{3} - 2\,n x^{4} + 20\,n^{2} x^{2} - 6\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} + 9\,x^{3} - 6\,x^{4} + x^{5} + 6\,n x^{2} - 9\,n x^{3} + 3\,n x^{4} - 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 11\,x - 8\,x^{2} + x^{3} + n - 13\,n x + 5\,n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 3\,x^{2} - x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21528
311726650
414519615424420
511051036210410441204410
6122944701905222940802872055440
7148117070134090322820262044692645040810810

All 15 rows

Technical details
Node id
n-fc743cefdb386d8c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-31e93cd5a554a770

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: 1,    1;
  2: 1,    5,      2;
  3: 1,   17,     26,       6;
  4: 1,   45,    196,     154,       24;
  5: 1,  105,   1036,    2104,     1044,      120;
  6: 1,  229,   4470,   19052,    22940,     8028,      720;
  7: 1,  481,  17070,  134090,   322820,   262044,    69264,     5040;
  8: 1,  989,  60344,  803150,  3415900,  5355844,  3173256,   663696,   40320;
  9: 1, 2009, 202784, 4317260, 29890000, 80405884, 89624136, 40897656, 6999840, 362880;
%F Formula
The row polynomials satisfy (n-2+x)*P(n,x) = -((4*n^3-20*n^2+31*n-15)*x^2+(6*n^2-21*n+18)*x^3+(2*n-3)*x^4)*P(n-2,x) - ((2*n^2-6*n+4)*x^2-(2*n^2-9*n+9)*x^3-(3*n-6)*x^4-x^5)*d/dx P(n-2,x) + (n-2+(4*n^2-13*n+11)*x+(5*n-8)*x^2+x^3)*P(n-1,x) + ((n-2)*x-(n-3)*x^2-x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^3 - 20*n^2 + 31*n - 15)*x^2 + (6*n^2 - 21*n + 18)*x^3 + (2*n - 3)*x^4)*P[n - 2, x] - ((2*n^2 - 6*n + 4)*x^2 - (2*n^2 - 9*n + 9)*x^3 - (3*n - 6)*x^4 - x^5)*D[P[n - 2, x], x] + (n - 2 + (4*n^2 - 13*n + 11)*x + (5*n - 8)*x^2 + x^3)*P[n - 1, x] + ((n - 2)*x - (n - 3)*x^2 - x^3)*D[P[n - 1, x], x])/(n - 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 + x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^3 - 20*n^2 + 31*n - 15)*x^2 + (6*n^2 - 21*n + 18)*x^3 + (2*n - 3)*x^4)*v[i - 2] - ((2*n^2 - 6*n + 4)*x^2 - (2*n^2 - 9*n + 9)*x^3 - (3*n - 6)*x^4 - x^5)*deriv(v[i - 2], x) + (n - 2 + (4*n^2 - 13*n + 11)*x + (5*n - 8)*x^2 + x^3)*v[i - 1] + ((n - 2)*x - (n - 3)*x^2 - x^3)*deriv(v[i - 1], x))/(n - 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

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.