Real-rooted atlas

A094816 → basis A013610 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(2 - n\right)\,P_{n-2}(x) + \left(3 + 3\,x + n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2538
32933971
419331216227694
5145729762205702817421
612341299072812512825283524386276
711612532121335994620911566420109357291084483
81203329370227647542953303153134095531808740824218714665106

All 15 rows

Technical details
Node id
n-388c3ec60ee31657 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-49e3f6dc0463a475

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:         5,         3;
   3:        29,        33,         9;
   4:       193,       312,       162,        27;
   5:      1457,      2976,      2205,       702,        81;
   6:     12341,     29907,     28125,     12825,      2835,       243;
   7:    116125,    321213,    359946,    209115,     66420,     10935,      729;
   8:   1203329,   3702276,   4754295,   3303153,   1340955,    318087,    40824,    2187;
   9:  13627073,  45788808,  65638746,  52436916,  25535979,   7763364,  1439046,  148716,   6561;
  10: 167525317, 606517515, 952635762, 852170922, 478550835, 174986973, 41671098, 6232950, 531441, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*P(n-2,x) + (n+3+3*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(n - 2)*P[n - 2, x] + (n + 3 + 3*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] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*v[i - 2] + (n + 3 + 3*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

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