Real-rooted atlas

A063967 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-3 - 10\,x - 8\,x^{2} + 3\,n + 6\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-3\,x - 8\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
3915630
427726016175
581297387216441025
62431134202517467321206000
772941319396110437110238832835125
8218714580403386026452704270727584896205625

All 15 rows

Technical details
Node id
n-c9b41f28c6b3a1b3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-87f1729e0e693bae

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 -1)
  • 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:     1;
   2:     3,      2;
   3:     9,     15,      6;
   4:    27,     72,     60,      16;
   5:    81,    297,    387,     216,      44;
   6:   243,   1134,   2025,    1746,     732,     120;
   7:   729,   4131,   9396,   11043,    7110,    2388,    328;
   8:  2187,  14580,  40338,   60264,   52704,   27072,   7584,    896;
   9:  6561,  50301, 164025,  297918,  330561,  230040,  98280,  23616,  2448;
  10: 19683, 170586, 640791, 1371978, 1848987, 1629882, 941544, 344304, 72432, 6688;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (3*n-3+(6*n-10)*x+(4*n-8)*x^2)*P(n-1,x) - (3*x+8*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (6*n - 10)*x + (4*n - 8)*x^2)*P[n - 1, x] - (3*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n - 3 + (6*n - 10)*x + (4*n - 8)*x^2)*v[i - 1] - (3*x + 8*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n - 1)); 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

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
  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