Real-rooted atlas

A078812 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 6\,x + 2\,x^{2} + 12\,n - 6\,n x - 2\,n x^{2} - 6\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(4\,n - 12\,n x - 7\,n x^{2} - n x^{3} + 3\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-10 - 37\,x - 37\,x^{2} - 11\,x^{3} - x^{4} - 5\,n - 6\,n x - n x^{2} + 5\,n^{2} + 6\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2415
3148123
44643121102
514519287161441
64447684921462011871
713312856239110002202417823
8392610085104885725177030928132332

All 15 rows

Technical details
Node id
n-2c3a50cd6c534a8f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ab0d65133ac4f36a

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     4,      1;
   3:    14,      8,      1;
   4:    46,     43,     12,      1;
   5:   145,    192,     87,     16,     1;
   6:   444,    768,    492,    146,    20,     1;
   7:  1331,   2856,   2391,   1000,   220,    24,    1;
   8:  3926,  10085,  10488,   5725,  1770,   309,   28,   1;
   9: 11434,  34264,  42731,  28968, 11670,  2856,  413,  32,  1;
  10: 32960, 112991, 164676, 133918, 66948, 21315, 4312, 532, 36, 1;
%F Formula
The row polynomials satisfy (2*n^3-6*n^2+12*n-8-(6*n-6)*x-(2*n-2)*x^2)*P(n,x) = (3*n^3+4*n+(n^3-12*n)*x-7*n*x^2-n*x^3)*P(n-1,x) + (5*n^2-5*n-10+(6*n^2-6*n-37)*x+(n^2-n-37)*x^2-11*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 + 4*n + (n^3 - 12*n)*x - 7*n*x^2 - n*x^3)*P[n - 1, x] + (5*n^2 - 5*n - 10 + (6*n^2 - 6*n - 37)*x + (n^2 - n - 37)*x^2 - 11*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 + 12*n - 8 - (6*n - 6)*x - (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n^3 + 4*n + (n^3 - 12*n)*x - 7*n*x^2 - n*x^3)*v[i - 1] + (5*n^2 - 5*n - 10 + (6*n^2 - 6*n - 37)*x + (n^2 - n - 37)*x^2 - 11*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 + 12*n - 8 - (6*n - 6)*x - (2*n - 2)*x^2)); 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

Same fingerprint ?

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