Real-rooted atlas

A257623 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(3 + 12\,x + 15\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
296666141
327582158410563249
481481826994443522217698421
524339030403920123024014414405765763691449
6729313698568029028607040589881605362156817873856165085341
721872513958772944486058113601957475520303509606422521058566434588168573758209

All 15 rows

Technical details
Node id
n-6a4d5848d20747a3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-04577fbe25f76a46

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:     3,         6;
  2:     9,        66,          66;
  3:    27,       582,        1584,         1056;
  4:    81,      4818,       26994,        44352,         22176;
  5:   243,     39030,      403920,      1230240,       1441440,        576576;
  6:   729,    313698,     5680290,     28607040,      58988160,      53621568,      17873856;
  7:  2187,   2513958,    77294448,    605811360,    1957475520,    3035096064,    2252105856,      643458816;
  8:  6561,  20124786,  1032481362,  12141315648,   57743975520,  135877053312,  168407471232,   105527245824,   26381811456;
  9: 19683, 161037654, 13643630352, 235063383456, 1583079065568, 5305900856256, 9769635203328, 10072704305664, 5461034971392, 1213563326976;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3+12*x+15*x^2+6*x^3)*P(n-1,x) + (5*x+15*x^2+15*x^3+5*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[((3 + 12*x + 15*x^2 + 6*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*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] = ((3 + 12*x + 15*x^2 + 6*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*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