Real-rooted atlas

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

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
28210
3401656
425615216424
5195215042723728
61740816608451227238800
7177280201216721927936458624
820316162675584119961620070479366115456
9258667523876300820842752467148835379290497792
10362283008608740864382025216107896064125992963537921473898240
11553526272010306482176739542630425221468163993098242236825626180996096
12916203765761872411586561512376934406051190784012163368960106371993622368256503860593664
13163316515635236343661445123263596400640149938290688036612378624043861757952190375731210442399909888
14311911597998087507843100672074202975117312385082160947201108294033408016972052398081163552153601903757312231879186565120
Technical details
Node id
n-f44b7b0b2818cec1 · 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 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;
  2:        8,        2;
  3:       40,       16;
  4:      256,      152,       16;
  5:     1952,     1504,      272;
  6:    17408,    16608,     4512,     272;
  7:   177280,   201216,    72192,    7936;
  8:  2031616,  2675584,  1199616,  200704,   7936;
  9: 25866752, 38763008, 20842752, 4671488, 353792;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+(n-1)*x)*P(n-1,x) - (4+6*x+2*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 + (n - 1)*x)*P[n - 1, x] - (4 + 6*x + 2*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 + (n - 1)*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 - 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