Real-rooted atlas

A060187 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
388117
44872261147
53847684648011697
63840960081602640242124483
74608013824015168072960141687281423857
864512022579203037440194880059572873752218618560947

All 15 rows

Technical details
Node id
n-b83dd8ae6ec13b4b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2a2e8db5a042c827

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:         8,         8,          1;
   4:        48,        72,         26,          1;
   5:       384,       768,        464,         80,         1;
   6:      3840,      9600,       8160,       2640,       242,         1;
   7:     46080,    138240,     151680,      72960,     14168,       728,        1;
   8:    645120,   2257920,    3037440,    1948800,    595728,     73752,     2186,       1;
   9:  10321920,  41287680,   65802240,   52899840,  22305024,   4612608,   377504,    6560,     1;
  10: 185794560, 836075520, 1540546560, 1490227200, 804948480, 237646080, 34553280, 1913760, 19682, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-2+(2*n-3)*x)*P(n-1,x) - (2*x+2*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*n - 2 + (2*n - 3)*x)*P[n - 1, x] - (2*x + 2*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*n - 2 + (2*n - 3)*x)*v[i - 1] - (2*x + 2*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