Real-rooted atlas

narayana → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
3107118
4353912187
512619698181439
64629306602002512278
7171642573960172536033112052
86435190192202212705385059542164669

All 15 rows

Technical details
Node id
n-b65a6aa89ef4c6a9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0c8c0fcad0285056

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:     3,      1;
   3:    10,      7,      1;
   4:    35,     39,     12,      1;
   5:   126,    196,     98,     18,      1;
   6:   462,    930,    660,    200,     25,     1;
   7:  1716,   4257,   3960,   1725,    360,    33,     1;
   8:  6435,  19019,  22022,  12705,   3850,   595,    42,    1;
   9: 24310,  83512, 116116,  84084,  34034,  7700,   924,   52,  1;
  10: 92378, 361998, 588744, 515424, 263718, 80262, 14196, 1368, 63, 1;
%F Formula
The row polynomials satisfy (n^3-n+(2*n^2-n-3)*x+(n+1)*x^2)*P(n,x) = (4*n^3-2*n^2-4*n+2+(3*n^3+4*n^2-16*n+7)*x+(6*n^2-8*n+4)*x^2+(3*n-3)*x^3)*P(n-1,x) - ((2*n^2+n-1)*x+(2*n^2+5*n-3)*x^2+4*n*x^3+2*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[((4*n^3 - 2*n^2 - 4*n + 2 + (3*n^3 + 4*n^2 - 16*n + 7)*x + (6*n^2 - 8*n + 4)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] - ((2*n^2 + n - 1)*x + (2*n^2 + 5*n - 3)*x^2 + 4*n*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^3 - n + (2*n^2 - n - 3)*x + (n + 1)*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] = ((4*n^3 - 2*n^2 - 4*n + 2 + (3*n^3 + 4*n^2 - 16*n + 7)*x + (6*n^2 - 8*n + 4)*x^2 + (3*n - 3)*x^3)*v[i - 1] - ((2*n^2 + n - 1)*x + (2*n^2 + 5*n - 3)*x^2 + 4*n*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^3 - n + (2*n^2 - n - 3)*x + (n + 1)*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