Real-rooted atlas

A167884 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 - 7\,x + 8\,n + 8\,n x\right)\,P_{n-1}(x) + \left(-8\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
22020141
336054018211083
4936018720110001640140721
53182407956006660002034001476211998003
61336608040098240435708002031120036054201328601121084601
766830400023390640003125435040196592760057267756062620740119574218735224683

All 15 rows

Technical details
Node id
n-b44f129e99b8f204 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b84a4525108a87c3

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:            20,             20,              1;
  3:           360,            540,            182,              1;
  4:          9360,          18720,          11000,           1640,              1;
  5:        318240,         795600,         666000,         203400,          14762,             1;
  6:      13366080,       40098240,       43570800,       20311200,        3605420,        132860,            1;
  7:     668304000,     2339064000,     3125435040,     1965927600,      572677560,      62620740,      1195742,           1;
  8:   38761632000,   155046528000,   245882407680,   194984375040,    79765227360,   15444112320,   1076510000,    10761680,        1;
  9: 2558267712000, 11512204704000, 21131772480000, 20237581728000, 10706377106880, 3033799423200, 405479988000, 18408286800, 96855122, 1;
%F Formula
The row polynomials satisfy P(n,x) = (8*n-6+(8*n-7)*x)*P(n-1,x) - (8*x+8*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[(8*n - 6 + (8*n - 7)*x)*P[n - 1, x] - (8*x + 8*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] = (8*n - 6 + (8*n - 7)*x)*v[i - 1] - (8*x + 8*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