Real-rooted atlas

A167884 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(8\,x + 24\,x^{2} + 24\,x^{3} + 8\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21202041
311825403601083
4116401100018720936040721
51147622034006660007956003182401998003
61132860360542020311200435708004009824013366080121084601
711195742626207405726775601965927600312543504023390640006683040008735224683

All 15 rows

Technical details
Node id
n-56a2efc225b761f1 · revision 0 · JSON
Definition source
derived:mobius_x_over_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: 1,        2;
  2: 1,       20,          20;
  3: 1,      182,         540,          360;
  4: 1,     1640,       11000,        18720,          9360;
  5: 1,    14762,      203400,       666000,        795600,         318240;
  6: 1,   132860,     3605420,     20311200,      43570800,       40098240,       13366080;
  7: 1,  1195742,    62620740,    572677560,    1965927600,     3125435040,     2339064000,      668304000;
  8: 1, 10761680,  1076510000,  15444112320,   79765227360,   194984375040,   245882407680,   155046528000,    38761632000;
  9: 1, 96855122, 18408286800, 405479988000, 3033799423200, 10706377106880, 20237581728000, 21131772480000, 11512204704000, 2558267712000;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (8*x+24*x^2+24*x^3+8*x^4)*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[Together[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (8*x + 24*x^2 + 24*x^3 + 8*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = ((1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (8*x + 24*x^2 + 24*x^3 + 8*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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