Real-rooted atlas

A091042 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-3 - 4\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(12\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
29413
3456016121
4189504336641093
57293240388817282569841
626731782033264253448448102488573
7947788452235872269568149760399364096797161
832805408240146966423328001900800829440184320163847174453

All 15 rows

Technical details
Node id
n-cda2c26e87214b18 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-72244d199a450cc9

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 -1)
  • 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:      1;
   2:      9,       4;
   3:     45,      60,       16;
   4:    189,     504,      336,        64;
   5:    729,    3240,     3888,      1728,       256;
   6:   2673,   17820,    33264,     25344,      8448,      1024;
   7:   9477,   88452,   235872,    269568,    149760,     39936,     4096;
   8:  32805,  408240,  1469664,   2332800,   1900800,    829440,   184320,    16384;
   9: 111537, 1784592,  8328096,  17449344,  19388160,  12220416,  4386816,   835584,   65536;
  10: 373977, 7479540, 43879968, 117013248, 169019136, 143410176, 73543680, 22413312, 3735552, 262144;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = (6*n-3+(4*n-4)*x)*P(n-1,x) + (12*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6*n - 3 + (4*n - 4)*x)*P[n - 1, x] + (12*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((6*n - 3 + (4*n - 4)*x)*v[i - 1] + (12*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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

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