Real-rooted atlas

A207606 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 64 + 69\,x + 18\,x^{2}\)
\(P_{1}(x) = 373 + 600\,x + 315\,x^{2} + 54\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0646918151
1373600315541342
2217446503672126916211927
3126713382235667185764779486106001
47385223629531179621718884240172531458942082
543044116055042543994221994011522253557526050743748372737
6250879410688052197686802073443413488930557393414288402077651312274412551
714622323700491901481322421814927041419846577355464225234335552873670202739366661340222
8852251444534740571079172342151261851613828971248617078893706410961086676562078524823422771180985877649447
949672854129064705847687323981121317354181269325363592453840644783735908175840048845001242976370040773541935429452237504801
10289514610218475535070537641832969433976593511117183094092690357400560679265442479294299679544907901805915250275404536253320211062882464259893762
11168741480711166321781423703243810057151994100809373570298108804122531566096948955563152030371201216557612993462259383070624375479778514408237335531886464126101539057
12983497423247319399779952518078637160530782983756076585685781608001854250966624333820626036955099663921667589188364570896931627145967002236270290892603426731568266251941956593836670653957751
13573224305873457005691680016933967375940386860155267606093754395183070106418987120608558970949024058787138001220970409268061839352656102425814303707505990976908281015652838331187451770485615145128697814325909784080702
1433409960929142840807444042411279589636888027761016545082047402474171310059544929152724456921130132451242239940959838624591848056003811270149023870040523194905336113118481679582403258043914375676705782407508958802740641237860934422896517402768567
Technical details
Node id
n-2b36967cabde26ca · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-958aa4aeaa0c1134

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:        64,         69,         18;
  1:       373,        600,        315,          54;
  2:      2174,       4650,       3672,        1269,         162;
  3:     12671,      33822,      35667,       18576,        4779,        486;
  4:     73852,     236295,     311796,      217188,       84240,      17253,       1458;
  5:    430441,    1605504,    2543994,     2219940,     1152225,     355752,      60507,       4374;
  6:   2508794,   10688052,   19768680,    20734434,    13488930,    5573934,    1428840,     207765,     13122;
  7:  14622323,   70049190,  148132242,   181492704,   141984657,   73554642,   25234335,    5528736,    702027,    39366;
  8:  85225144,  453474057, 1079172342,  1512618516,  1382897124,  861707889,  370641096,  108667656,  20785248,  2342277,  118098;
  9: 496728541, 2906470584, 7687323981, 12131735418, 12693253635, 9245384064, 4783735908, 1758400488, 450012429, 76370040, 7735419, 354294;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 64 + 69*x + 18*x^2, P(1,x) = 373 + 600*x + 315*x^2 + 54*x^3.
%t Mathematica
P[0, x_] := 64 + 69*x + 18*x^2;
P[1, x_] := 373 + 600*x + 315*x^2 + 54*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 3*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] = 64 + 69*x + 18*x^2; v[2] = 373 + 600*x + 315*x^2 + 54*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (6 + 3*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