Real-rooted atlas

A085478 → D → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-4\,n - n x + 3\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(-5 - 6\,x - x^{2} + 5\,n + 6\,n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
10
211
3527
41916338
56584334186
6210366222565859
7654143411824608563821
81985524454752900825120716556
9591118268231091554060151344161870356
1017345613829125474532366451113020442089294549
115030520056834272632972819708576020189562952261101218611
1214451664093212377761371620966770453432143514303124095320114993298
13411900201126043320005435500441775024444009405902520004612555003851220297422
141166209621664814775114207108801908085512181068553005618006484177806743071944561381954351
Technical details
Node id
n-c6d513a1d8c60449 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-992190218dce3d0f

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -2)
  • 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:    0;
  1:    0;
  2:    1;
  3:    5,     2;
  4:   19,    16,     3;
  5:   65,    84,    33,     4;
  6:  210,   366,   222,    56,    5;
  7:  654,  1434,  1182,   460,   85,    6;
  8: 1985,  5244,  5475,  2900,  825,  120,   7;
  9: 5911, 18268, 23109, 15540, 6015, 1344, 161, 8;
%F Formula
The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (3*n^2-4*n+(n^2-n)*x)*P(n-1,x) + (5*n-5+(6*n-6)*x+(n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 4*n + (n^2 - n)*x)*P[n - 1, x] + (5*n - 5 + (6*n - 6)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((3*n^2 - 4*n + (n^2 - n)*x)*v[i - 1] + (5*n - 5 + (6*n - 6)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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