Real-rooted atlas

A125662 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-4\,x - 3\,x^{2} + 8\,n + 12\,n x + 3\,n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} - 3\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
216321563
36419218456496
4256102414889282093905
51024512099849472436478030744
6409624576601607680053856196482911242047
71638411468833792054272051264028454485872108641905632

All 15 rows

Technical details
Node id
n-dac117530db899a6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-cf9b351a67dba233

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:      4,       4;
  2:     16,      32,      15;
  3:     64,     192,     184,       56;
  4:    256,    1024,    1488,      928,      209;
  5:   1024,    5120,    9984,     9472,     4364,      780;
  6:   4096,   24576,   60160,    76800,    53856,    19648,     2911;
  7:  16384,  114688,  337920,   542720,   512640,   284544,    85872,   10864;
  8:  65536,  524288, 1806336,  3497984,  4161280,  3111936,  1427808,  367296,   40545;
  9: 262144, 2359296, 9306112, 21102592, 30299136, 28550144, 17646336, 6895872, 1545492, 151316;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (8*n+(12*n-4)*x+(3*n-3)*x^2)*P(n-1,x) - (4*x+8*x^2+3*x^3)*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[((8*n + (12*n - 4)*x + (3*n - 3)*x^2)*P[n - 1, x] - (4*x + 8*x^2 + 3*x^3)*D[P[n - 1, x], x])/(2*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] = ((8*n + (12*n - 4)*x + (3*n - 3)*x^2)*v[i - 1] - (4*x + 8*x^2 + 3*x^3)*deriv(v[i - 1], x))/(2*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

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