Real-rooted atlas

A078812 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-16 + 8\,n\right)\,P_{n}(x) = \left(12 + 16\,x + 12\,n + 16\,n x\right)\,P_{n-1}(x) + \left(5 + 24\,x + 16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
244
3243256
4100288192580
53601632230410245320
6120074881651215360512045680
7381630432921601331209216024576376264
8117561143364425608755209369605160961146883011916
9354004066241921536485376070963206021120275251252428823611560
101047801388736776179223933952446208005160960036241408141557762359296182176140
1130600045978242969625610834432024643584036298752034681651220761804870778880104857601388066960
1288406414852736108913920459479040123640832022202449922698149888219414528011442585603460300804613734410469504224
1325316644703289638616192018509670405760184320122325073921810877644818709217280132356505606113198080166094438420132659278308498576
14719650014649926413317692167152952320252989798406212616192010871321395213671648460812275417088076818677760318347673607851737088872415232581625025940
15203290804500368964488141312267106590721059015168002955810816005979529543688884149288969702510428167699169280004318455398401622692331523664143974437580963844294201927960
Technical details
Node id
n-3e9edb6397a669eb · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6d3434c9512a4a8b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      0;
   2:      4;
   3:     24,      32;
   4:    100,     288,     192;
   5:    360,    1632,    2304,     1024;
   6:   1200,    7488,   16512,    15360,     5120;
   7:   3816,   30432,   92160,   133120,    92160,    24576;
   8:  11756,  114336,  442560,   875520,   936960,   516096,   114688;
   9:  35400,  406624, 1921536,  4853760,  7096320,  6021120,  2752512,   524288;
  10: 104780, 1388736, 7761792, 23933952, 44620800, 51609600, 36241408, 14155776, 2359296;
%F Formula
The row polynomials satisfy (8*n-16)*P(n,x) = (12*n+12+(16*n+16)*x)*P(n-1,x) + (5+24*x+16*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 12 + (16*n + 16)*x)*P[n - 1, x] + (5 + 24*x + 16*x^2)*D[P[n - 1, x], x])/(8*n - 16)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((12*n + 12 + (16*n + 16)*x)*v[i - 1] + (5 + 24*x + 16*x^2)*deriv(v[i - 1], x))/(8*n - 16)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023