Real-rooted atlas

A142963 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + n x\right)\,P_{n-1}(x) + \left(4\,x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
2410115
3872301111
4164645167411071
532286470162584166112663
664173768448064240107403541177255
712810470494905613217604502804000873412866671

All 15 rows

Technical details
Node id
n-9ea4f5e2e9e2537e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6df2e759b56c31d7

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,      10,         1;
  3:   8,      72,        30,         1;
  4:  16,     464,       516,        74,         1;
  5:  32,    2864,      7016,      2584,       166,         1;
  6:  64,   17376,     84480,     64240,     10740,       354,        1;
  7: 128,  104704,    949056,   1321760,    450280,     40008,      734,      1;
  8: 256,  629248,  10223488,  24198976,  14713840,   2681296,   139108,   1498,    1;
  9: 512, 3777792, 107268864, 410350080, 410042976, 132557712, 14341992, 462264, 3030, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+n*x)*P(n-1,x) + (4*x-x^2)*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)*P[n - 1, x] + (4*x - x^2)*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)*v[i - 1] + (4*x - 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 - 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