Real-rooted atlas

A094816 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 24\,x - 9\,n + 8\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(4 + 8\,x - 16\,x^{2} - 8\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 8\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)
\(P_{3}(x) = 25 + 52\,x + 16\,x^{2}\)
\(P_{4}(x) = 108 + 380\,x + 336\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
325521693
410838033664888
55292544372819202568977
6300217716353602976010240102497102
7196731340043298243772802099205222440961127021
8147200111529231950244521216341376013619202580481638414028844

All 15 rows

Technical details
Node id
n-991bbb9b271e4162 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-984f99a16dacc37c

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:        6,         4;
   3:       25,        52,        16;
   4:      108,       380,       336,        64;
   5:      529,      2544,      3728,      1920,       256;
   6:     3002,     17716,     35360,     29760,     10240,      1024;
   7:    19673,    134004,    329824,    377280,    209920,     52224,      4096;
   8:   147200,   1115292,   3195024,   4521216,   3413760,   1361920,    258048,    16384;
   9:  1241889,  10222400,  32871200,  54435584,  50959104,  27553792,   8314880,  1245184,   65536;
  10: 11679694, 102798372, 362221632, 677439104, 745016832, 500247552, 204374016, 48463872, 5898240, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(8*n-24)*x)*P(n-3,x) - (n^2-4+(8*n-8)*x+16*x^2)*P(n-2,x) + (2*n-1+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 6 + 4*x, P(3,x) = 25 + 52*x + 16*x^2, P(4,x) = 108 + 380*x + 336*x^2 + 64*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[3, x_] := 25 + 52*x + 16*x^2;
P[4, x_] := 108 + 380*x + 336*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*P[n - 3, x] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*P[n - 2, x] + (2*n - 1 + 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] = 6 + 4*x; v[3] = 25 + 52*x + 16*x^2; v[4] = 108 + 380*x + 336*x^2 + 64*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*v[i - 3] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*v[i - 2] + (2*n - 1 + 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023