Real-rooted atlas

A207606 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 89 + 324\,x + 288\,x^{2}\)
\(P_{1}(x) = 610 + 3312\,x + 5904\,x^{2} + 3456\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
089324288701
161033125904345613282
2418130180807849504041472251657
328657258120921744163123214307844976644768201
419641821205449468864223845122954880020653056597196890344162
513462691694270490806832268686720474024960498659328289640448716636161711770877
692274651326336128294914082948104512651286656091582617608005423104397733068885996339232433302501
7632459861022222160730723636830321941760804932951041417635717121656474624001238347284485374771200010319560704614520976642

All 15 rows

Technical details
Node id
n-29982f17f6b1d2a0 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-11954e259cdef45c

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         89,         324,          288;
  1:        610,        3312,         5904,          3456;
  2:       4181,       30180,        80784,         95040,         41472;
  3:      28657,      258120,       921744,       1631232,       1430784,         497664;
  4:     196418,     2120544,      9468864,      22384512,      29548800,       20653056,        5971968;
  5:    1346269,    16942704,     90806832,     268686720,     474024960,      498659328,      289640448,       71663616;
  6:    9227465,   132633612,    829491408,    2948104512,    6512866560,     9158261760,     8005423104,     3977330688,      859963392;
  7:   63245986,  1022222160,   7307236368,   30321941760,   80493295104,   141763571712,   165647462400,   123834728448,    53747712000,    10319560704;
  8:  433494437,  7781873340,  62587829088,  296992324224,  920803500288,  1949106281472,  2852689674240,  2850635317248,  1861390761984,   717209468928,  123834728448;
  9: 2971215073, 58652824464, 524190047328, 2799678276864, 9929039097600, 24551622402048, 43192455634944, 54062888583168, 47183611428864, 27346835865600, 9473356726272, 1486016741376;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 89 + 324*x + 288*x^2, P(1,x) = 610 + 3312*x + 5904*x^2 + 3456*x^3.
%t Mathematica
P[0, x_] := 89 + 324*x + 288*x^2;
P[1, x_] := 610 + 3312*x + 5904*x^2 + 3456*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 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] = 89 + 324*x + 288*x^2; v[2] = 610 + 3312*x + 5904*x^2 + 3456*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 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