Real-rooted atlas

A086645 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3416828
48729632208
5162566405121281552
63280032004480256051211584
764230413440286722764812288204886464
8128627250176150528215040157696573448192645376

All 15 rows

Technical details
Node id
n-180b0ec0ce696564 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d44a70ca53ce7ac3

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:   2,     2;
   3:   4,    16,      8;
   4:   8,    72,     96,      32;
   5:  16,   256,    640,     512,     128;
   6:  32,   800,   3200,    4480,    2560,      512;
   7:  64,  2304,  13440,   28672,   27648,    12288,    2048;
   8: 128,  6272,  50176,  150528,  215040,   157696,   57344,    8192;
   9: 256, 16384, 172032,  688128, 1351680,  1441792,  851968,  262144,   32768;
  10: 512, 41472, 552960, 2838528, 7299072, 10543104, 8945664, 4423680, 1179648, 131072;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) + (4*x+2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] + (4*x + 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] + (4*x + 2*x^2)*deriv(v[i - 1], x))/(n - 2)); 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