Real-rooted atlas

A086645 → basis A013611 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 64\,x - 64\,x^{2}\right)\,P_{n-2}(x) + \left(12 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26814
35612864248
4576192019205124928
560162662440960245764096102272
662976348160727040696320286720327682153984
76594564374528115507201526988810321920314572826214445584384
8690585653444608170655744288096256272498688139460608330301442097152966189056

All 15 rows

Technical details
Node id
n-f6b16b0bcda6a25a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-b860ced141a23493

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:         6,          8;
   3:        56,        128,          64;
   4:       576,       1920,        1920,         512;
   5:      6016,      26624,       40960,       24576,         4096;
   6:     62976,     348160,      727040,      696320,       286720,        32768;
   7:    659456,    4374528,    11550720,    15269888,     10321920,      3145728,      262144;
   8:   6905856,   53444608,   170655744,   288096256,    272498688,    139460608,    33030144,     2097152;
   9:  72318976,  639631360,  2395996160,  4924112896,   5997854720,   4345298944,  1761607680,   335544320,   16777216;
  10: 757334016, 7535591424, 32393134080, 78473330688, 117039955968, 109999816704, 63770198016, 21139292160, 3321888768, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = -(16+64*x+64*x^2)*P(n-2,x) + (12+16*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(16 + 64*x + 64*x^2)*P[n - 2, x] + (12 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 8*x; for(n = 3, N, my(i = n); v[i] = -(16 + 64*x + 64*x^2)*v[i - 2] + (12 + 16*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, v[i])))};
T(10)

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