Real-rooted atlas

A078812 → x ↦ x+1 → basis A013610

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
31524948
45613810827329
5209696837432812255
6780327353284212162024315456
72911147363029432400190355832729105937
8108646440416005621627017172080190204122187726103
940545275472803142131284813166558300883214896998465614976784
1015131611591193878928744454890334447190127375580812422162361961968334111385
115647194814952181899274010212857150765550307523627212416166304466487178732059049233802911
1221075601979484683325636207533745339876000384385176306424944172239372669222001712421025981561771471602508992
137865521806871123744971551040009760192495447025021379522342583180159206601677974204526847612061706205850305653144110983760033
1429354524326550165165672432050759967601047997116015399030042165703216321322377423278282440283396006405104965502421895369227634932159432375283811239
15109552575131357712072320506202423415024055222920450905338956961101357934741010139958087020455676336800281584143249331064016276784766869363892820884782969516002918640
Technical details
Node id
n-6db6c305c50929d0 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-5f3ddac121e398b5

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:      4,       3;
   3:     15,      24,       9;
   4:     56,     138,     108,      27;
   5:    209,     696,     837,     432,      81;
   6:    780,    3273,    5328,    4212,    1620,     243;
   7:   2911,   14736,   30294,   32400,   19035,    5832,     729;
   8:  10864,   64404,  160056,  216270,  171720,   80190,   20412,    2187;
   9:  40545,  275472,  803142, 1312848, 1316655,  830088,  321489,   69984,   6561;
  10: 151316, 1159119, 3878928, 7444548, 9033444, 7190127, 3755808, 1242216, 236196, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+3*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 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] = 4 + 3*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (4 + 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

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