Real-rooted atlas

A094816 → x ↦ x+1 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 - n\right)\,P_{n-2}(x) + \left(1 + 4\,x + n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
311281655
44917619264481
52611168180811522564645
61631851616560150406400102449171
71174368812157504180800110080337924096566827
895901615184159342421669761675520738304172032163847073725
987680960537601729244826777856246525441384857646448648519686553695064361
108877691651771242016845763463904003648852482450380801051115522782003241287682621441369375615
11986410117631524122525291376472242240055423116804275360768219968716874661888016023552019660800104857621054430591
12119355623396637741603386455177668028752704872908889607529855590444646260736182265937925028249600893911040922746884194304344231563897
1315624736141131672386640484980616368103591409856014332220277761356314370048902045315072425544548352141539278848324114841604855955456427819008167772165964569413645
14220048367319192161945424473950241707681666228891142424594496143360251740209909761842018207744097726303600643764928380928104160244531220137797222425790775296196293427267108864109196040092491
1533176523072712991436371086011970211653648028270983598457644152920757836848353023361843238369240810700822451073494220897489364910080313040033546247325329915904121260264652813435194572889254789122684354562106381399472435
Technical details
Node id
n-2e08ede65b06f117 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-78aab4e9b3808cb9

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:       3,        4;
   3:      11,       28,        16;
   4:      49,      176,       192,        64;
   5:     261,     1168,      1808,      1152,       256;
   6:    1631,     8516,     16560,     15040,      6400,      1024;
   7:   11743,    68812,    157504,    180800,    110080,     33792,      4096;
   8:   95901,   615184,   1593424,   2166976,   1675520,    738304,    172032,    16384;
   9:  876809,  6053760,  17292448,  26777856,  24652544,  13848576,   4644864,   851968,   65536;
  10: 8877691, 65177124, 201684576, 346390400, 364885248, 245038080, 105111552, 27820032, 4128768, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*P(n-2,x) + (n+1+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n - 2)*P[n - 2, x] + (n + 1 + 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] = 3 + 4*x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*v[i - 2] + (n + 1 + 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