Real-rooted atlas

A103327 → basis A013609 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 24\,x - 36\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16612
2449636176
3328112810802162752
42448115201814410368129643776
518272109152240192233280907207776699392
61363849876482794176385689626179207464964665611186176
710179848662656299669765375808052617600267805445878656279936178962432

All 15 rows

Technical details
Node id
n-26895ab1507a1876 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f93791dbc19bff2a

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:        6,         6;
  2:       44,        96,         36;
  3:      328,      1128,       1080,        216;
  4:     2448,     11520,      18144,      10368,        1296;
  5:    18272,    109152,     240192,     233280,       90720,        7776;
  6:   136384,    987648,    2794176,    3856896,     2617920,      746496,      46656;
  7:  1017984,   8662656,   29966976,   53758080,    52617600,    26780544,    5878656,     279936;
  8:  7598336,  74293248,  303897600,  671625216,   862410240,   640991232,  256048128,   44789760,   1679616;
  9: 56714752, 626443776, 2958280704, 7773677568, 12379309056, 12171617280, 7179798528, 2324588544, 332563968, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = -(4+24*x+36*x^2)*P(n-2,x) + (8+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(4 + 24*x + 36*x^2)*P[n - 2, x] + (8 + 12*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] = 6 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 24*x + 36*x^2)*v[i - 2] + (8 + 12*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

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