Real-rooted atlas

A391045 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-2 + 8\,n - 10\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(2\,n + 7\,n x - 6\,n^{2} - 16\,n^{2} x + 4\,n^{3} + 7\,n^{3} x\right)\,P_{n-1}(x) + \left(-10\,x + 10\,x^{2} + 10\,n x - 18\,n x^{2} + 12\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x) + \left(-12\,x^{2} + 3\,x^{3} + 20\,n x^{2} - 5\,n x^{3}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
338112
443018153
5580126321244
661755603605011152
77336189024008257215531
885885292115507700163898126875

All 15 rows

Technical details
Node id
n-998d2f929b97c6f3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c6c82bb93c30543f

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:  2,    1;
   3:  3,    8,     1;
   4:  4,   30,    18,      1;
   5:  5,   80,   126,     32,      1;
   6:  6,  175,   560,    360,     50,      1;
   7:  7,  336,  1890,   2400,    825,     72,     1;
   8:  8,  588,  5292,  11550,   7700,   1638,    98,    1;
   9:  9,  960, 12936,  44352,  50050,  20384,  2940,  128,   1;
  10: 10, 1485, 28512, 144144, 252252, 171990, 47040, 4896, 162, 1;
%F Formula
The row polynomials satisfy (4*n^3-10*n^2+8*n-2)*P(n,x) = (4*n^3-6*n^2+2*n+(7*n^3-16*n^2+7*n)*x)*P(n-1,x) + ((12*n^2+10*n-10)*x+(2*n^2-18*n+10)*x^2)*d/dx P(n-1,x) + ((20*n-12)*x^2-(5*n-3)*x^3)*d^2/dx^2 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[((4*n^3 - 6*n^2 + 2*n + (7*n^3 - 16*n^2 + 7*n)*x)*P[n - 1, x] + ((12*n^2 + 10*n - 10)*x + (2*n^2 - 18*n + 10)*x^2)*D[P[n - 1, x], x] + ((20*n - 12)*x^2 - (5*n - 3)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 10*n^2 + 8*n - 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] = ((4*n^3 - 6*n^2 + 2*n + (7*n^3 - 16*n^2 + 7*n)*x)*v[i - 1] + ((12*n^2 + 10*n - 10)*x + (2*n^2 - 18*n + 10)*x^2)*deriv(v[i - 1], x) + ((20*n - 12)*x^2 - (5*n - 3)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 10*n^2 + 8*n - 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. [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