Real-rooted atlas

A142459 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-6 - 7\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
31212125
4120180621363
516803360199231217345
63024075600636001980015621190803
7665280199584021756001024800187572781216056905
817297280605404808107344051332400151981201735020390621227215803

All 15 rows

Technical details
Node id
n-8bbd0ca4e55b3927 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d6ac9abcd3bcb1be

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:          12,          12,            1;
   4:         120,         180,           62,            1;
   5:        1680,        3360,         1992,          312,           1;
   6:       30240,       75600,        63600,        19800,        1562,           1;
   7:      665280,     1995840,      2175600,      1024800,      187572,        7812,          1;
   8:    17297280,    60540480,     81073440,     51332400,    15198120,     1735020,      39062,         1;
   9:   518918400,  2075673600,   3297127680,   2626525440,  1085424480,   214925760,   15849552,    195312,      1;
  10: 17643225600, 79394515200, 145919854080, 140211751680, 74694674880, 21461176800, 2952530400, 143817840, 976562, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-6+(4*n-7)*x)*P(n-1,x) - (4*x+4*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[(4*n - 6 + (4*n - 7)*x)*P[n - 1, x] - (4*x + 4*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] = (4*n - 6 + (4*n - 7)*x)*v[i - 1] - (4*x + 4*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