Real-rooted atlas

A008292 → γ (scale 1/4) → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3Pn(1)
111
222
38210
4401656
525615216424
6195215042723728
71740816608451227238800
8177280201216721927936458624

All 15 rows

Technical details
Node id
n-c7584eaab51b7a88 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c71c852095bde979

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
   3:        8,        2;
   4:       40,       16;
   5:      256,      152,       16;
   6:     1952,     1504,      272;
   7:    17408,    16608,     4512,     272;
   8:   177280,   201216,    72192,    7936;
   9:  2031616,  2675584,  1199616,  200704,   7936;
  10: 25866752, 38763008, 20842752, 4671488, 353792;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-2+(n-2)*x)*P(n-1,x) - (4+6*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 + (n - 2)*x)*P[n - 1, x] - (4 + 6*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 + (n - 2)*x)*v[i - 1] - (4 + 6*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