Real-rooted atlas

A046716 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(18 + 48\,x + 32\,x^{2} - 9\,n - 24\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-2 - 4\,x + 3\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2448
319361671
411832427264778
5102137604784230425612125
6120885758810329683392266241024284012
7183727109256425965763084992183654444339240969241891
83389242243303407268536011566771210342118449283072980172816384378595022
973156249617016576222799241644646778885362098944385956249615416606722637168646553618409947641
10180434954817507570148742577902081798207599362719089223682628925962241587047628805469251174482378752002621441029827400400
11500091798195491285640682678016836400761301950860813902475644672169122758277121370486923673671335216087042164107313152291533750272104857664998958518719
1215379206545261886204988550810404051537467233986845482643272814699846656010690365098895361089222611136512760479696568320348194225782784944049537679361150939771699241943044565303338264082
1351952155415381704149885866096433598224000155216011745563124480393976115726300166782094648165580883346530500333568731195694441431044487665847487692818350620151906304449943759539404850114102873292816777216353016345110857429
141911990785926432283874849801053001930911568719392007956499238778821762212017462411335680437109663908987904062952276565029355526657719434647584768513153007975609139228111232729963560961038924818725994496232575937990885376238495285955788806710886429793105387299603252
15761372016112369991229138729020475988915480816958247961641653387645551332416129166077145235963648288272584749755410432476369818440736772096590148820444757311488548088128717574045696376886574758868025344186510497844183433216629011681605425561601295754323432990310412313362650271580162684354562724646021507044539675
Technical details
Node id
n-31d1c82323726714 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-88b89a1b70ed720b

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:          4,           4;
   3:         19,          36,          16;
   4:        118,         324,         272,           64;
   5:       1021,        3760,        4784,         2304,          256;
   6:      12088,       57588,      103296,        83392,        26624,         1024;
   7:     183727,     1092564,     2596576,      3084992,      1836544,       443392,         4096;
   8:    3389242,    24330340,    72685360,    115667712,    103421184,     49283072,      9801728,       16384;
   9:   73156249,   617016576,  2227992416,   4464677888,   5362098944,   3859562496,   1541660672,   263716864,      65536;
  10: 1804349548, 17507570148, 74257790208, 179820759936, 271908922368, 262892596224, 158704762880, 54692511744, 8237875200, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n-18+(24*n-48)*x+(16*n-32)*x^2)*P(n-2,x) + (3*n-2+(4*n-4)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9*n - 18 + (24*n - 48)*x + (16*n - 32)*x^2)*P[n - 2, x] + (3*n - 2 + (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] = 4 + 4*x; for(n = 3, N, my(i = n); v[i] = -(9*n - 18 + (24*n - 48)*x + (16*n - 32)*x^2)*v[i - 2] + (3*n - 2 + (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