Real-rooted atlas

A172431 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8\,n - 2\,n x + 8\,n x^{2} + 8\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(2 - 4\,x - 14\,x^{2} - 8\,x^{3} - 42\,n - 30\,n x + 34\,n x^{2} + 24\,n x^{3} + 48\,n^{2} + 96\,n^{2} x + 48\,n^{2} x^{2} + 32\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(-15 - 62\,x - 95\,x^{2} - 64\,x^{3} - 16\,x^{4} - 50\,n - 130\,n x - 112\,n x^{2} - 32\,n x^{3} - 20\,n^{2} - 36\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
21816337
35982344179
4185352231605833
5564136812425169563791
616854980580833681001140716989
7495717302247261854077501764196875243
814406580409837390828493751591228982649330105
941455189446371858408740274795115750300244512345101436935
10118321604964135050117242481384705724164246631530646732440116213781
113354001897488474944469129846465938405174017193124876968900197025501226719267
129451935863080162674402660120028427750207987201060479037601209080501430001358567613114333617
132650229178881385450142098955720119024500996987605948838025410360769333516087502216501856481914487160639
147398330539858561792105833578154004786034504519830003093463801549610405665657514894100273344533321624843980152067947213
Technical details
Node id
n-b286006c6ec70e71 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d250e7e45ce67209

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:     5,      2;
  2:    18,     16,      3;
  3:    59,     82,     34,      4;
  4:   185,    352,    231,     60,      5;
  5:   564,   1368,   1242,    516,     95,      6;
  6:  1685,   4980,   5808,   3368,   1001,    140,     7;
  7:  4957,  17302,  24726,  18540,   7750,   1764,   196,    8;
  8: 14406,  58040,  98373,  90828,  49375,  15912,  2898,  264,   9;
  9: 41455, 189446, 371858, 408740, 274795, 115750, 30024, 4512, 345, 10;
%F Formula
The row polynomials satisfy (8*n^3+8*n^2-8*n+(16*n^2-2*n)*x+8*n*x^2)*P(n,x) = (32*n^3+48*n^2-42*n+2+(24*n^3+96*n^2-30*n-4)*x+(48*n^2+34*n-14)*x^2+(24*n-8)*x^3)*P(n-1,x) - (20*n^2+50*n+15+(36*n^2+130*n+62)*x+(16*n^2+112*n+95)*x^2+(32*n+64)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((32*n^3 + 48*n^2 - 42*n + 2 + (24*n^3 + 96*n^2 - 30*n - 4)*x + (48*n^2 + 34*n - 14)*x^2 + (24*n - 8)*x^3)*P[n - 1, x] - (20*n^2 + 50*n + 15 + (36*n^2 + 130*n + 62)*x + (16*n^2 + 112*n + 95)*x^2 + (32*n + 64)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 + 8*n^2 - 8*n + (16*n^2 - 2*n)*x + 8*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((32*n^3 + 48*n^2 - 42*n + 2 + (24*n^3 + 96*n^2 - 30*n - 4)*x + (48*n^2 + 34*n - 14)*x^2 + (24*n - 8)*x^3)*v[i - 1] - (20*n^2 + 50*n + 15 + (36*n^2 + 130*n + 62)*x + (16*n^2 + 112*n + 95)*x^2 + (32*n + 64)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 + 8*n^2 - 8*n + (16*n^2 - 2*n)*x + 8*n*x^2)); 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. [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