Real-rooted atlas

A111125 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(24 - 28\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(-8\,n - 16\,n x + 12\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-5 - 24\,x - 16\,x^{2} + 5\,n + 24\,n x + 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
244
3283260
4124320192636
54601920249610245900
6156091201881616384512051000
75016379201086721484809728024576421944
815572144768534720100864010291205406721146883388180
9471565209602364096572928080332806537216286720052428826623476
101401801795360968332828787712517171205763072038993920146800642359296205787700
1141078059865603745804813227827229105664041459712038305792022177382473138176104857601570243100
12119006419450560138610176567823360148284416025832325123044966400240176332812150374403565158404613734411857571184
1334157286188563249507584023104460806996592640144527523842080692633620903362560143799091206459228160170708172820132659288778002800
14972816419353216017179311369003919360310591641607435866931212682199040015542570188813598982144082931875840334957117448053063680872415232659933524516
1527525580596536160581991052833863611392131200496640357707243520706666168320102513141350410930052136968467356057604636803072001701209702403751385497637580963844875826953900
Technical details
Node id
n-d665d4c0a624e188 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b2309e6cedfb5a0a

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:     28,      32;
   4:    124,     320,     192;
   5:    460,    1920,    2496,     1024;
   6:   1560,    9120,   18816,    16384,     5120;
   7:   5016,   37920,  108672,   148480,    97280,    24576;
   8:  15572,  144768,  534720,  1008640,  1029120,   540672,   114688;
   9:  47156,  520960, 2364096,  5729280,  8033280,  6537216,  2867200,   524288;
  10: 140180, 1795360, 9683328, 28787712, 51717120, 57630720, 38993920, 14680064, 2359296;
%F Formula
The row polynomials satisfy (8*n^2-28*n+24)*P(n,x) = (12*n^2-8*n+(16*n^2-16*n)*x)*P(n-1,x) + (5*n-5+(24*n-24)*x+(16*n-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^2 - 8*n + (16*n^2 - 16*n)*x)*P[n - 1, x] + (5*n - 5 + (24*n - 24)*x + (16*n - 16)*x^2)*D[P[n - 1, x], x])/(8*n^2 - 28*n + 24)]];
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^2 - 8*n + (16*n^2 - 16*n)*x)*v[i - 1] + (5*n - 5 + (24*n - 24)*x + (16*n - 16)*x^2)*deriv(v[i - 1], x))/(8*n^2 - 28*n + 24)); 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