Real-rooted atlas

A225117 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
3113418
4160938162
512511131545161944
611018106781449829333229160
714089889982624381541131517764524880
81163766924993870352489028714757367710112811022480

All 15 rows

Technical details
Node id
n-89455cdf5080c940 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-08a5fbe77f597ff1

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: 1,      2;
   3: 1,     13,        4;
   4: 1,     60,       93,         8;
   5: 1,    251,     1131,       545,         16;
   6: 1,   1018,    10678,     14498,       2933,         32;
   7: 1,   4089,    88998,    262438,     154113,      15177,         64;
   8: 1,  16376,   692499,   3870352,    4890287,    1475736,      77101,       128;
   9: 1,  65527,  5175013,  50476003,  117758659,   77404933,   13270807,    388321,     256;
  10: 1, 262134, 37732212, 608260290, 2388954618, 2887100154, 1103599596, 114709518, 1948005, 512;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n-4)*x)*P(n-1,x) + (3*x-3*x^2)*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[(1 + (3*n - 4)*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], x]];
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] = (1 + (3*n - 4)*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], 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

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