Real-rooted atlas

A079638 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
214372475
3903534541921089
4744384673496166192020025
57560482741217711519549393623040446535
691440692532216198435662213281914159914432256011715795
7128520011229756416635328517760910372939675303388301935845160960353743425

All 15 rows

Technical details
Node id
n-11124d93b13aca24 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cbe3741e35d7b661

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,          4;
  2:        14,         37,          24;
  3:        90,        353,         454,         192;
  4:       744,       3846,        7349,        6166,         1920;
  5:      7560,      48274,      121771,      151954,        93936,        23040;
  6:     91440,     692532,     2161984,     3566221,      3281914,      1599144,      322560;
  7:   1285200,   11229756,    41663532,    85177609,    103729396,     75303388,    30193584,     5160960;
  8:  20603520,  203598000,   873158012,  2124586164,   3210395649,   3086846596,  1845321372,   627345072,    92897280;
  9: 371226240, 4086822096, 19857223188, 55930492988, 100701362091, 120252606648, 95283096868, 48324697056, 14239656000, 1857945600;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(2*n^2-2*n+(6*n^2-6*n)*x+(6*n^2-6*n)*x^2+(2*n^2-2*n)*x^3)*P(n-2,x) + (3*n+(6*n+1)*x+(3*n+1)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 2*n + (6*n^2 - 6*n)*x + (6*n^2 - 6*n)*x^2 + (2*n^2 - 2*n)*x^3)*P[n - 2, x] + (3*n + (6*n + 1)*x + (3*n + 1)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 2*n + (6*n^2 - 6*n)*x + (6*n^2 - 6*n)*x^2 + (2*n^2 - 2*n)*x^3)*v[i - 2] + (3*n + (6*n + 1)*x + (3*n + 1)*x^2)*v[i - 1])/(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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023