Real-rooted atlas

A086645 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 96\,x - 144\,x^{2}\right)\,P_{n-2}(x) + \left(12 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
261218
356192144392
45762880432017289504
5601639936921608294420736241792
6629765222401635840235008014515202488326271488
76594566561792259891205153587252254720238878722985984163874816
869058568016691238397542497232486413795246081059028992376233984358318084293992448

All 15 rows

Technical details
Node id
n-fafe18af774a8f28 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-2b7ed1e86e9f26f3

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,          12;
   3:        56,         192,         144;
   4:       576,        2880,        4320,         1728;
   5:      6016,       39936,       92160,        82944,        20736;
   6:     62976,      522240,     1635840,      2350080,      1451520,       248832;
   7:    659456,     6561792,    25989120,     51535872,     52254720,     23887872,      2985984;
   8:   6905856,    80166912,   383975424,    972324864,   1379524608,   1059028992,    376233984,     35831808;
   9:  72318976,   959447040,  5390991360,  16618881024,  30364139520,  32997113856,  20065812480,   5733089280,   429981696;
  10: 757334016, 11303387136, 72884551680, 264847491072, 592514777088, 835311108096, 726382411776, 361184624640, 85136375808, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = -(16+96*x+144*x^2)*P(n-2,x) + (12+24*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(16 + 96*x + 144*x^2)*P[n - 2, x] + (12 + 24*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 + 12*x; for(n = 3, N, my(i = n); v[i] = -(16 + 96*x + 144*x^2)*v[i - 2] + (12 + 24*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