Real-rooted atlas

A063967 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 6\,x - 5\,n - 3\,n x + n^{2}\right)\,P_{n}(x) = \left(-48 - 60\,x - 18\,x^{2} + 22\,n + 15\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-16 - 44\,x - 36\,x^{2} - 9\,x^{3} + 4\,n + 8\,n x + 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
233
3151833
46014481285
52167748913242205
6732349260754536121516050
72388142203312937584206554374112350
87584541441581122410562016908748015309765375

All 15 rows

Technical details
Node id
n-ed3bd3d067300fee · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-32abd0dfa14f0fb6

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:     3;
   3:    15,     18;
   4:    60,    144,      81;
   5:   216,    774,     891,     324;
   6:   732,   3492,    6075,    4536,    1215;
   7:  2388,  14220,   33129,   37584,   20655,    4374;
   8:  7584,  54144,  158112,  241056,  201690,   87480,   15309;
   9: 23616, 196560,  690120, 1322244, 1489590,  984150,  352107,   52488;
  10: 72432, 688608, 2824632, 6519528, 9244935, 8231868, 4485537, 1364688, 177147;
%F Formula
The row polynomials satisfy (n^2-5*n+6-(3*n-6)*x)*P(n,x) = -(2*n^2-22*n+48-(15*n-60)*x+18*x^2)*P(n-1,x) + (4*n-16+(8*n-44)*x+(3*n-36)*x^2-9*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 22*n + 48 - (15*n - 60)*x + 18*x^2)*P[n - 1, x] + (4*n - 16 + (8*n - 44)*x + (3*n - 36)*x^2 - 9*x^3)*D[P[n - 1, x], x])/(n^2 - 5*n + 6 - (3*n - 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 22*n + 48 - (15*n - 60)*x + 18*x^2)*v[i - 1] + (4*n - 16 + (8*n - 44)*x + (3*n - 36)*x^2 - 9*x^3)*deriv(v[i - 1], x))/(n^2 - 5*n + 6 - (3*n - 6)*x)); 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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.