Real-rooted atlas

narayana → D → D

nonnegative coefficients

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

Recurrence

\(\left(6 + n - 4\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(-6\,x - n + 17\,n x - 14\,n^{2} x + n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} - 4\,n x + 4\,n x^{2} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 2\)

Rows ?

nx0x1x2x3x4x5Pn(1)
10
20
322
412618
5406012112
610030018020600
721010501260420302970
83922940588039208404214014

All 15 rows

Technical details
Node id
n-1dcf17454d9bc598 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-fcecb9b7df4a0d68

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 -3)
  • 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:    0;
   2:    0;
   3:    2;
   4:   12,     6;
   5:   40,    60,    12;
   6:  100,   300,   180,     20;
   7:  210,  1050,  1260,    420,    30;
   8:  392,  2940,  5880,   3920,   840,    42;
   9:  672,  7056, 21168,  23520, 10080,  1512,   56;
  10: 1080, 15120, 63504, 105840, 75600, 22680, 2520, 72;
%F Formula
The row polynomials satisfy (n^3-4*n^2+n+6)*P(n,x) = (n^3-n+(3*n^3-14*n^2+17*n-6)*x)*P(n-1,x) + ((2*n^2-4*n+2)*x-(2*n^2-4*n+2)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n + (3*n^3 - 14*n^2 + 17*n - 6)*x)*P[n - 1, x] + ((2*n^2 - 4*n + 2)*x - (2*n^2 - 4*n + 2)*x^2)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((n^3 - n + (3*n^3 - 14*n^2 + 17*n - 6)*x)*v[i - 1] + ((2*n^2 - 4*n + 2)*x - (2*n^2 - 4*n + 2)*x^2)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + n + 6)); 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