Real-rooted atlas

A290595 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 - 11\,x + 6\,n x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
3419427
4814722228405
516887412831942808505
632483552538113566550243640229635
76425167555684256267632687881107336582407577955
8128128203530115045415640117517960100544412255261921106560295540245

All 15 rows

Technical details
Node id
n-ede94e2c84d7b416 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-bc5ca063cc7eafc5

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:   2,       1;
   3:   4,      19,         4;
   4:   8,     147,       222,         28;
   5:  16,     887,      4128,       3194,         280;
   6:  32,    4835,     52538,     113566,       55024,         3640;
   7:  64,   25167,    555684,    2562676,     3268788,      1107336,        58240;
   8: 128,  128203,   5301150,   45415640,   117517960,    100544412,     25526192,      1106560;
   9: 256,  646519,  47537320,  695714590,  3189383200,   5352311764,   3325767376,    663605680,    24344320;
  10: 512, 3245139, 410038434, 9696965250, 72479948400, 208996478388, 248493947496, 118361719296, 19213911360, 608608000;
%F Formula
The row polynomials satisfy P(n,x) = (2+(6*n-11)*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[(2 + (6*n - 11)*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] = (2 + (6*n - 11)*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