Real-rooted atlas

A046716 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3410519
48445016118
516224428288651021
632150440884216192232612088
76413472464646376843376146261957183727
81281522566238721056144945568472524125050137003389242

All 15 rows

Technical details
Node id
n-77a956978a91074c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-7319b1f17529ae73

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,       10,         5;
   4:   8,       44,        50,        16;
   5:  16,      224,       428,       288,        65;
   6:  32,     1504,      4088,      4216,      1922,       326;
   7:  64,    13472,     46464,     63768,     43376,     14626,      1957;
   8: 128,   152256,    623872,   1056144,    945568,    472524,    125050,    13700;
   9: 256,  2058240,   9673600,  19332352,  21309104,  14000064,   5485560,  1187472,   109601;
  10: 512, 32174592, 169870080, 389739776, 507841632, 411264096, 212053808, 68009904, 12408738, 986410;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n-8+(4*n-8)*x+(n-2)*x^2)*P(n-2,x) + (2*n-2+n*x)*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[-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*P[n - 2, x] + (2*n - 2 + n*x)*P[n - 1, x]];
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] = -(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*v[i - 2] + (2*n - 2 + n*x)*v[i - 1]); 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