Real-rooted atlas

A008459 → D → D

nonnegative coefficients

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

Recurrence

\(\left(4 - 4\,n + n^{2}\right)\,P_{n}(x) = \left(6\,x - 10\,n x + n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5Pn(1)
00
10
222
318624
4729612180
5200600300201120
645024002700720306300
7882735014700882014704233264

All 15 rows

Technical details
Node id
n-eced3c738adffaba · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5f98bd4aea48ed7d

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:    2;
  3:   18,     6;
  4:   72,    96,     12;
  5:  200,   600,    300,     20;
  6:  450,  2400,   2700,    720,     30;
  7:  882,  7350,  14700,   8820,   1470,    42;
  8: 1568, 18816,  58800,  62720,  23520,  2688,   56;
  9: 2592, 42336, 190512, 317520, 211680, 54432, 4536, 72;
%F Formula
The row polynomials satisfy (n^2-4*n+4)*P(n,x) = (n^2+(3*n^2-10*n+6)*x)*P(n-1,x) + ((2*n-2)*x-(2*n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + (3*n^2 - 10*n + 6)*x)*P[n - 1, x] + ((2*n - 2)*x - (2*n - 2)*x^2)*D[P[n - 1, x], x])/(n^2 - 4*n + 4)]];
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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((n^2 + (3*n^2 - 10*n + 6)*x)*v[i - 1] + ((2*n - 2)*x - (2*n - 2)*x^2)*deriv(v[i - 1], x))/(n^2 - 4*n + 4)); 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