Real-rooted atlas

narayana → θ+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
336110
441812135
554060201126
6675200150301462
771265257003154211716
881961176245019605885616435

All 15 rows

Technical details
Node id
n-f41df8cc2ea6fcca · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4b23e46a501a23b5

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,   6,    1;
   4:  4,  18,   12,     1;
   5:  5,  40,   60,    20,     1;
   6:  6,  75,  200,   150,    30,     1;
   7:  7, 126,  525,   700,   315,    42,     1;
   8:  8, 196, 1176,  2450,  1960,   588,    56,    1;
   9:  9, 288, 2352,  7056,  8820,  4704,  1008,   72,  1;
  10: 10, 405, 4320, 17640, 31752, 26460, 10080, 1620, 90, 1;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (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(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((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 - n)]];
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] = ((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 - n)); 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