Real-rooted atlas

A046716 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 24\,x + 92\,x^{2} + 120\,x^{3} + 50\,x^{4} - n - 12\,n x - 46\,n x^{2} - 60\,n x^{3} - 25\,n x^{4}\right)\,P_{n-2}(x) + \left(-4\,x - 8\,x^{2} - 4\,x^{3} + n + 7\,n x + 11\,n x^{2} + 5\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2268
35304176
416148424356944
56583638307204440116336
632654743550010962015707078826386816
7195740994349119153582036367554300402189348111758528
8137003460003678396212460247153894013898624014167226056341416433822976

All 15 rows

Technical details
Node id
n-d26ed5f0b4a8a0cb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-69ca13488ba57d9b

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,        6;
   3:      5,       30,        41;
   4:     16,      148,       424,        356;
   5:     65,      836,      3830,       7204,        4401;
   6:    326,     5474,     35500,     109620,      157070,       78826;
   7:   1957,    40994,    349119,    1535820,     3636755,     4300402,      1893481;
   8:  13700,   346000,   3678396,   21246024,    71538940,   138986240,    141672260,     56341416;
   9: 109601,  3251752,  41635676,  299665432,  1320876934,  3630528536,   6022771548,   5430521384,   1978638881;
  10: 986410, 33695166, 506090184, 4378455544, 23985706572, 85981414404, 200671962856, 291770944920, 236748607194, 79749105326;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n-2+(12*n-24)*x+(46*n-92)*x^2+(60*n-120)*x^3+(25*n-50)*x^4)*P(n-2,x) + (n+(7*n-4)*x+(11*n-8)*x^2+(5*n-4)*x^3)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (12*n - 24)*x + (46*n - 92)*x^2 + (60*n - 120)*x^3 + (25*n - 50)*x^4)*P[n - 2, x] + (n + (7*n - 4)*x + (11*n - 8)*x^2 + (5*n - 4)*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (12*n - 24)*x + (46*n - 92)*x^2 + (60*n - 120)*x^3 + (25*n - 50)*x^4)*v[i - 2] + (n + (7*n - 4)*x + (11*n - 8)*x^2 + (5*n - 4)*x^3)*v[i - 1])/(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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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