Real-rooted atlas

A091044 → basis A013611 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 4\,x + 16\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-10 - 18\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112820
212816048336
31344243214082565440
414080332802841610240128087296
51474564300804853762621446656061441397760
6154419253575687520256542310420930564014082867222368256
716171008650444801092157449863168051281920151388162293760131072357908480
8169345024774766592151447142416441671681077084160431751168101842944125829125898245726601216
91773404160909377536020284178432257320550402035417088010336337920333447168064801996866846720262144091625881600
101857132953610549723136026442674995238389206220835602825216021918803558490025492480240836935683944742912346030080115343361466015154176
111944814878721212288073728337317671731255158117498885869836173312425005468876821240291000327267554426881648487301122316094668817532190725033164823456246661120
12203663658188813821070540800422725445222407688327004160092352702054400769171600179204533790965760018940134686720551802634240010796138496001319528038408724152320218103808375299963355136
1321327935176704156525922353152521963348951040104518834454528013989044027392001317738916085760895488601620480442387000197120157859174154240398257553408006813562961920732828794880427483463689395240966004799480791040
142233490368102401762625050378240636454641126604813913976052121600205340910026752002158800886824960016623375640166400948802299101184040136268054528001243889637785600275421488742400416665514803203981703118848206695301120402653184096076791961092096

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-16 - 32\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(12 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12 + 8\,x\)
Technical details
Node id
n-a10f777f75d58fc1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-748291d881f1ad3e)
Fingerprint
f6-2cb32fb53e87d6dc

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:         12,          8;
  2:        128,        160,          48;
  3:       1344,       2432,        1408,         256;
  4:      14080,      33280,       28416,       10240,        1280;
  5:     147456,     430080,      485376,      262144,       66560,        6144;
  6:    1544192,    5357568,     7520256,     5423104,     2093056,      401408,      28672;
  7:   16171008,   65044480,   109215744,    98631680,    51281920,    15138816,    2293760,    131072;
  8:  169345024,  774766592,  1514471424,  1644167168,  1077084160,   431751168,  101842944,  12582912,   589824;
  9: 1773404160, 9093775360, 20284178432, 25732055040, 20354170880, 10336337920, 3334471680, 648019968, 66846720, 2621440;
%F Formula
The row polynomials satisfy n*P(n,x) = (16*n-4+(12*n-4)*x)*P(n-1,x) - (10+18*x+8*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 4 + (12*n - 4)*x)*P[n - 1, x] - (10 + 18*x + 8*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n - 4 + (12*n - 4)*x)*v[i - 1] - (10 + 18*x + 8*x^2)*deriv(v[i - 1], x))/(n)); 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

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