Real-rooted atlas

A111125 → D → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3729
43120354
5115120394278
63905702946451323
71254237016985809566003
838939048835539401005132726380

All 15 rows

Technical details
Node id
n-2bfe40bbed4239a0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f0aff1f976ed95f5

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     0;
   2:     1;
   3:     7,      2;
   4:    31,     20,      3;
   5:   115,    120,     39,      4;
   6:   390,    570,    294,     64,     5;
   7:  1254,   2370,   1698,    580,    95,     6;
   8:  3893,   9048,   8355,   3940,  1005,   132,    7;
   9: 11789,  32560,  36939,  22380,  7845,  1596,  175,   8;
  10: 35045, 112210, 151302, 112452, 50505, 14070, 2380, 224, 9;
%F Formula
The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (3*n^2-2*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(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 2*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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((3*n^2 - 2*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, 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