Real-rooted atlas

narayana → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(-18 + 14\,x - 2\,x^{2} + n + 3\,n x - n x^{2} + 12\,n^{2} - 9\,n^{2} x + n^{2} x^{2} - 6\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(16\,x - 12\,x^{2} + 2\,x^{3} + 5\,n - 61\,n x + 47\,n x^{2} - 7\,n x^{3} + 2\,n^{2} + 46\,n^{2} x - 31\,n^{2} x^{2} + 3\,n^{2} x^{3} - 2\,n^{3} - 18\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-6\,x + 11\,x^{2} - 6\,x^{3} + x^{4} + 14\,n x - 26\,n x^{2} + 14\,n x^{3} - 2\,n x^{4} - 8\,n^{2} x + 12\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3527
41818339
550100424196
6115400330805930
72311260178584013564257
84203332735058801820210719019

All 15 rows

Technical details
Node id
n-d2ed48e052463ed7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9f01c67d4f0b38b0

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    0;
   2:    1;
   3:    5,     2;
   4:   18,    18,     3;
   5:   50,   100,    42,      4;
   6:  115,   400,   330,     80,      5;
   7:  231,  1260,  1785,    840,    135,     6;
   8:  420,  3332,  7350,   5880,   1820,   210,    7;
   9:  708,  7728, 24696,  30576,  15960,  3528,  308,   8;
  10: 1125, 16200, 71064, 127008, 102060, 37800, 6300, 432, 9;
%F Formula
The row polynomials satisfy (n^4-6*n^3+12*n^2+n-18+(2*n^3-9*n^2+3*n+14)*x+(n^2-n-2)*x^2)*P(n,x) = (n^4-2*n^3+2*n^2+5*n+(3*n^4-18*n^3+46*n^2-61*n+16)*x+(6*n^3-31*n^2+47*n-12)*x^2+(3*n^2-7*n+2)*x^3)*P(n-1,x) + ((2*n^3-8*n^2+14*n-6)*x-(2*n^3-12*n^2+26*n-11)*x^2-(4*n^2-14*n+6)*x^3-(2*n-1)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 - 2*n^3 + 2*n^2 + 5*n + (3*n^4 - 18*n^3 + 46*n^2 - 61*n + 16)*x + (6*n^3 - 31*n^2 + 47*n - 12)*x^2 + (3*n^2 - 7*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 - 8*n^2 + 14*n - 6)*x - (2*n^3 - 12*n^2 + 26*n - 11)*x^2 - (4*n^2 - 14*n + 6)*x^3 - (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 6*n^3 + 12*n^2 + n - 18 + (2*n^3 - 9*n^2 + 3*n + 14)*x + (n^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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^4 - 2*n^3 + 2*n^2 + 5*n + (3*n^4 - 18*n^3 + 46*n^2 - 61*n + 16)*x + (6*n^3 - 31*n^2 + 47*n - 12)*x^2 + (3*n^2 - 7*n + 2)*x^3)*v[i - 1] + ((2*n^3 - 8*n^2 + 14*n - 6)*x - (2*n^3 - 12*n^2 + 26*n - 11)*x^2 - (4*n^2 - 14*n + 6)*x^3 - (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 6*n^3 + 12*n^2 + n - 18 + (2*n^3 - 9*n^2 + 3*n + 14)*x + (n^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. [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