Real-rooted atlas

A078812 → basis A013611 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(7 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 7 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
27815
34811264224
4329116013445123345
52255106401862414336409649951
6154569136021414424883214336032768745920
7105937752528221126434406402990080137625626214411138849
8726103602383221284928415257604831232033521664128450562097152166336815

All 15 rows

Technical details
Node id
n-36eb2fa2de362ed0 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-7e8ebe3586b974d9

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:        7,         8;
   3:       48,       112,         64;
   4:      329,      1160,       1344,        512;
   5:     2255,     10640,      18624,      14336,       4096;
   6:    15456,     91360,     214144,     248832,     143360,      32768;
   7:   105937,    752528,    2211264,    3440640,    2990080,    1376256,     262144;
   8:   726103,   6023832,   21284928,   41525760,   48312320,   33521664,   12845056,    2097152;
   9:  4976784,  47223120,  194973888,  457519104,  667402240,  619773952,  357826560,  117440512,   16777216;
  10: 34111385, 364352280, 1721317248, 4720899072, 8283656192, 9644113920, 7450132480, 3682598912, 1056964608, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+8*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 7 + 8*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 8*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] = 7 + 8*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (7 + 8*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