Real-rooted atlas

A008459 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2628
33024357
4140180604384
5630112063012052505
62772630050401680210616008
71201233264346501680037803367100849

All 15 rows

Technical details
Node id
n-b64160da1ac0b39d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-883c9e310e304e0b

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      1;
  2:      6,      2;
  3:     30,     24,       3;
  4:    140,    180,      60,       4;
  5:    630,   1120,     630,     120,      5;
  6:   2772,   6300,    5040,    1680,    210,      6;
  7:  12012,  33264,   34650,   16800,   3780,    336,     7;
  8:  51480, 168168,  216216,  138600,  46200,   7560,   504,   8;
  9: 218790, 823680, 1261260, 1009008, 450450, 110880, 13860, 720, 9;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (4*n^2-6*n+2+(3*n^2-6*n+2)*x)*P(n-1,x) - ((2*n-1)*x+(2*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (3*n^2 - 6*n + 2)*x)*P[n - 1, x] - ((2*n - 1)*x + (2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 6*n + 2 + (3*n^2 - 6*n + 2)*x)*v[i - 1] - ((2*n - 1)*x + (2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023