Real-rooted atlas

A116414 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3\right)\,P_{n-2}(x) + \left(7 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2465616118
3301564336641265
4196949844560179225613561
51288041072508483059289601024145376
6842533240725065444121601843204300840961558453
7551131248230046895524819520291200010352642007041638416706855
836051581860840841236432512583683910912018765824553369691750465536179100046
9235827131372325883490200005092957444700613122846914561131970562850816041287682621441919979941
101542635179991337442868361056480737510452102848003816787968191454412864959283214273740818350080104857620582479213
11100909648071992925122302800236843597182720542913100804670458060828329369600121198018563585146880698613760807403524194304220647331520
1266008848095143403227218138810345638287016320053879704704053264694067237938027724819620731289673147023360191758336003356491776352321536167772162365373209081
13431789042233648438876041406368846176327485200806451401860518405743603031040470126159462428546128936961286102974464424723087360999565557761587963494415267266566710886425357163305331
1428244967513425723207333041075979316328027400828951552474643192540165916802460262454745102454784381987146956802020173132595280599460085762388518764544509926703104741552947206576668672268435456271832676731398
Technical details
Node id
n-380dc94ffefbbbf3 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-09c3cedbcc4a7218

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        7,         4;
  2:       46,        56,        16;
  3:      301,       564,       336,        64;
  4:     1969,      4984,      4560,      1792,       256;
  5:    12880,     41072,     50848,     30592,      8960,      1024;
  6:    84253,    324072,    506544,    412160,    184320,     43008,      4096;
  7:   551131,   2482300,   4689552,   4819520,   2912000,   1035264,    200704,    16384;
  8:  3605158,  18608408,  41236432,  51258368,  39109120,  18765824,   5533696,   917504,   65536;
  9: 23582713, 137232588, 349020000, 509295744, 470061312, 284691456, 113197056, 28508160, 4128768, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -3*P(n-2,x) + (7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[-3*P[n - 2, x] + (7 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -3*v[i - 2] + (7 + 4*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 - 1, v[i])))};
T(9)

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