Real-rooted atlas

A185410 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1011
20235
304181537
40884180105377
5016360150021009454921
603214881080027300283501039578365
70646048722402940005292004365901351351473277
801282438446368028576807938000110602807567560202702531938737
902569792029040002610720010509912022004136024972948014594580034459425784384561
100512392448179136002291856001285956000379658664063567504006081075000310134825065472907521523937525
11010241571328109443840195888000014922230400597045556801383501319201926484560001592025435007202019825013749310575652667322517
12020486288384664519680164371468801668522240008805992025602713061070720515762727480061406695350004465252291500181490899590031623414322521672312694697
130409625159680401856000013614881280018164565619201240256489472049430644623360123217952457600198211715301600205977766995000133698296031300493325263431007905853580625782133969325801
14081921006510082423715840011173204224001938990493440016881180091776085326236835840027129469086720005662516067208000788557793063040072668956195836004259041440954300143886535167375021345804667687530481907097849485
Technical details
Node id
n-dba95db7d17b96cd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f2b1ecb4ea267ced

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: 0,   1;
  2: 0,   2,     3;
  3: 0,   4,    18,      15;
  4: 0,   8,    84,     180,      105;
  5: 0,  16,   360,    1500,     2100,       945;
  6: 0,  32,  1488,   10800,    27300,     28350,     10395;
  7: 0,  64,  6048,   72240,   294000,    529200,    436590,    135135;
  8: 0, 128, 24384,  463680,  2857680,   7938000,  11060280,   7567560,   2027025;
  9: 0, 256, 97920, 2904000, 26107200, 105099120, 220041360, 249729480, 145945800, 34459425;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (x+2*x^2+x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*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[((x + 2*x^2 + x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*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] = ((x + 2*x^2 + x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*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