Real-rooted atlas

A046716 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(50 + 80\,x + 32\,x^{2} - 25\,n - 40\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-4 - 4\,x + 5\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
26410
3415216109
4356644368641432
54401104008624281625626497
678826237060269600139200291841024754894
7189348170604841053696078768002960384455680409630787885
856341416252717652472122416470257920263457024787742729859072163841603546156
9197863888110398589664234110105282926892902421946280704986951680024651284482639790086553699602138593
107974910532648099334074012687498791041911688497536179960474880010838097623044077964328968764637184082390548482621447128277455538
1136220106234012466863191922074650083817200131735238806400149401468967168112922459716608568820641996801841377034240034760209858562915389931521048576576063289419661
1218289531857895613886638775905644743479835284720959939605057120012745256202771968116006578789836807330574049759232317554223587328090249732856217615195200002457611509420785664419430451832424202980320
13101605615118810418510991337273235232398769111280961673983528440939904011260443150038932481199436703553511424912373486758088704495609382774571008188409156343300096477384977353277447255713530249216501141129396224167772165136461847251936929
146157284642104619065641214565023081284236841845958314706886025317610828102278410344832448606527180812627612243199888384011237278152304429465673454976047131017216350042125335938990081185952855337834905627116097019020574723756731108729815042384952903178649667108864555721381650431686582
15404145384675814570014020628142801084821801845818570585371765984517766691226127095865699009407105537543677441362946515227750527180813895484657763067850752106231889677817106432006090054208424471887872258568790449151279104079028665758512578560016465748474676536934420961228963113336832123133626690620620826843545665167921144448534609677
Technical details
Node id
n-3731aa64809eacd4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cbfbc3a220f51650

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,            4;
   3:          41,           52,            16;
   4:         356,          644,           368,            64;
   5:        4401,        10400,          8624,          2816,           256;
   6:       78826,       237060,        269600,        139200,         29184,          1024;
   7:     1893481,      7060484,      10536960,       7876800,       2960384,        455680,         4096;
   8:    56341416,    252717652,     472122416,     470257920,     263457024,      78774272,      9859072,       16384;
   9:  1978638881,  10398589664,   23411010528,   29268929024,   21946280704,    9869516800,   2465128448,   263979008,      65536;
  10: 79749105326, 480993340740, 1268749879104, 1911688497536, 1799604748800, 1083809762304, 407796432896, 87646371840, 8239054848, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(25*n-50+(40*n-80)*x+(16*n-32)*x^2)*P(n-2,x) + (5*n-4+(4*n-4)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(25*n - 50 + (40*n - 80)*x + (16*n - 32)*x^2)*P[n - 2, x] + (5*n - 4 + (4*n - 4)*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 + 4*x; for(n = 3, N, my(i = n); v[i] = -(25*n - 50 + (40*n - 80)*x + (16*n - 32)*x^2)*v[i - 2] + (5*n - 4 + (4*n - 4)*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

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