Real-rooted atlas

A207607 → basis A007318 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
21913234
37169212163
4265334151292781
598915329172653723742
6369167835049194841145217929
713775292912606212576355558953285903
85140912415612849074418263855866799612411586
919186151874261205441358617640349260900910416921972027
107160352142673283846821919801092813367577844971311213157729448549
112672279876798512884491111928026386829251386169655613590418303162185245270718
129973081356015465746748155463719356473051607469652155881227060207801247101959932216905041
133722004514361128025258697926812955519166611097432228362404929987924203996130502032461232910121039254487
14138907099576063619109649171512696414809991466905653203262371786087496512817939967323533143290541684273110924979367394
Technical details
Node id
n-4f6f91e416b4e92b · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-57c034ccac65b14f

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:      5,      2;
  2:     19,     13,      2;
  3:     71,     69,     21,      2;
  4:    265,    334,    151,     29,      2;
  5:    989,   1532,    917,    265,     37,     2;
  6:   3691,   6783,   5049,   1948,    411,    45,    2;
  7:  13775,  29291,  26062,  12576,   3555,   589,   53,    2;
  8:  51409, 124156, 128490,  74418,  26385,  5866,  799,   61,  2;
  9: 191861, 518742, 612054, 413586, 176403, 49260, 9009, 1041, 69, 2;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + x)*P[n - 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] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + x)*v[i - 1]); 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

Candidate equalities ?

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