Real-rooted atlas

A172431 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(6\,n\right)\,P_{n}(x) = \left(-24 - 18\,x + 78\,n + 54\,n x\right)\,P_{n-1}(x) + \left(-68 - 99\,x - 36\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
2658427176
34418345221081905
4292972486669270040519951
519305588787117242552125551458205920
61268814594686867995415122370065443251032111201
7833049349113062124306075540351945012043082245321749621577935

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-16 - 24\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(9 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 6\,x\)
Technical details
Node id
n-c5c844748814fac4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-3aa08c307538a360)
Fingerprint
f6-e8b61b01ab78a634

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:        9,         6;
  2:       65,        84,        27;
  3:      441,       834,       522,       108;
  4:     2929,      7248,      6669,      2700,       405;
  5:    19305,     58878,     71172,     42552,     12555,      1458;
  6:   126881,    459468,    686799,    541512,    237006,     54432,      5103;
  7:   833049,   3491130,   6212430,   6075540,   3519450,   1204308,    224532,    17496;
  8:  5467345,  26021832,  53700705,  62671860,  45158715,  20522808,   5725566,   892296,   59049;
  9: 35877321, 191149302, 448753896, 608523840, 524423565, 297241974, 110495988, 25876584, 3444525, 196830;
%F Formula
The row polynomials satisfy 6*n*P(n,x) = (78*n-24+(54*n-18)*x)*P(n-1,x) - (68+99*x+36*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((78*n - 24 + (54*n - 18)*x)*P[n - 1, x] - (68 + 99*x + 36*x^2)*D[P[n - 1, x], x])/(6*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((78*n - 24 + (54*n - 18)*x)*v[i - 1] - (68 + 99*x + 36*x^2)*deriv(v[i - 1], x))/(6*n)); 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

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