Real-rooted atlas

A257617 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(7\,x + 21\,x^{2} + 21\,x^{3} + 7\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
24444492
3841211887922400
416374023540396001980086696
53233724417780133452015840006336004003656
664303644705544438214000808830007413120024710400225297752
712827330521162271881005919992338184000053311104003978374400113667840014952883560

All 15 rows

Technical details
Node id
n-a0a57351127b100c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8c4e3f9383454e05

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,         4;
  2:   4,        44,          44;
  3:   8,       412,        1188,          792;
  4:  16,      3740,       23540,        39600,         19800;
  5:  32,     33724,      417780,      1334520,       1584000,         633600;
  6:  64,    303644,     7055444,     38214000,      80883000,       74131200,       24710400;
  7: 128,   2733052,   116227188,   1005919992,    3381840000,     5331110400,     3978374400,     1136678400;
  8: 256,  24597980,  1889698580,  25228249200,  126603199800,   305469964800,   382961779200,   240975820800,    60243955200;
  9: 512, 221382844, 30505755060, 614264306040, 4428802224000, 15353691091200, 28763646912000, 29906008704000, 16265867904000, 3614637312000;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+8*x+10*x^2+4*x^3)*P(n-1,x) + (7*x+21*x^2+21*x^3+7*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[((2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x] + (7*x + 21*x^2 + 21*x^3 + 7*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] = ((2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1] + (7*x + 21*x^2 + 21*x^3 + 7*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