Real-rooted atlas

A122848 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 - x^{2} + n x + n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
246212
382418454
416801085610270
532240520460180261458
664672216029201920612768424
71281792806415680154008064215623251516

All 15 rows

Technical details
Node id
n-997d0b42ad183d9c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d941c5a22e857f2d

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:   2,     1;
  2:   4,     6,     2;
  3:   8,    24,    18,      4;
  4:  16,    80,   108,     56,     10;
  5:  32,   240,   520,    460,    180,     26;
  6:  64,   672,  2160,   2920,   1920,    612,     76;
  7: 128,  1792,  8064,  15680,  15400,   8064,   2156,    232;
  8: 256,  4608, 27776,  74368, 102480,  78848,  34384,   7968,   764;
  9: 512, 11520, 89856, 319872, 594720, 632016, 399840, 149328, 30456, 2620;
%F Formula
The row polynomials satisfy P(n,x) = (2+n*x+(n-1)*x^2)*P(n-1,x) - (2*x^2+x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + n*x + (n - 1)*x^2)*P[n - 1, x] - (2*x^2 + x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + n*x + (n - 1)*x^2)*v[i - 1] - (2*x^2 + x^3)*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 - 1, v[i])))};
T(9)

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