Real-rooted atlas

A084534 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-8 - 6\,x + 8\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
278217
3183015265
44710480242257
51233403551703521025
6322106814109323154824097
7843326252084396207953263216385
822079760182801878411440414484080265537
95778287466178574838558092622676021260992262145
1015127836202028402829802498151444885469013080181512021048577
11396032408126507051027466104813572041233718310593021384253014324194305
1210368268776820485323611528418072233304081858892725208193347335283432168216777217
1327144319506506349200123619601601145014508520941135043927001460030336050507654550195267108865
147106475499704194211504138792059305960602450804455066024227400966283527827805605607462059152242268435457

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 + 4\,x + 2\,x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-6 - 4\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 7 + 8\,x + 2\,x^{2}\)
Technical details
Node id
n-d9956ec4f2e74f86 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-a6999a1e90ac081b)
Fingerprint
f6-f83cf33d2c000fbc

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 0)
  • 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:    1;
  1:    3,     2;
  2:    7,     8,     2;
  3:   18,    30,    15,     2;
  4:   47,   104,    80,    24,     2;
  5:  123,   340,   355,   170,    35,     2;
  6:  322,  1068,  1410,   932,   315,    48,    2;
  7:  843,  3262,  5208,  4396,  2079,   532,   63,    2;
  8: 2207,  9760, 18280, 18784, 11440,  4144,  840,   80,  2;
  9: 5778, 28746, 61785, 74838, 55809, 26226, 7602, 1260, 99, 2;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (8*n-8+(6*n-6)*x)*P(n-1,x) - (5+9*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 8 + (6*n - 6)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n - 8 + (6*n - 6)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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

Same fingerprint ?

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