Real-rooted atlas

narayana → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 - n + n^{2}\right)\,P_{n}(x) = \left(2 + 6\,x - 6\,n - 21\,n x + 4\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} - 2\,n x - 6\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
233
3151833
46316281306
5252100811343242718
6990540097206480121523805
73861267306682572900328054374207495
815015126126405405623700467775153090153091806420

All 15 rows

Technical details
Node id
n-2326ef739163beb6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-dae6101d606a19f7

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:      3;
   3:     15,      18;
   4:     63,     162,       81;
   5:    252,    1008,     1134,      324;
   6:    990,    5400,     9720,     6480,     1215;
   7:   3861,   26730,    66825,    72900,    32805,     4374;
   8:  15015,  126126,   405405,   623700,   467775,   153090,    15309;
   9:  58344,  576576,  2270268,  4540536,  4864860,  2694384,   673596,   52488;
  10: 226746, 2577744, 12029472, 29719872, 41793570, 33434856, 14329224, 2834352, 177147;
%F Formula
The row polynomials satisfy (n^2-n-2)*P(n,x) = (4*n^2-6*n+2+(9*n^2-21*n+6)*x)*P(n-1,x) - ((2*n-1)*x+(6*n-3)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (9*n^2 - 21*n + 6)*x)*P[n - 1, x] - ((2*n - 1)*x + (6*n - 3)*x^2)*D[P[n - 1, x], x])/(n^2 - n - 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] = 3; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 6*n + 2 + (9*n^2 - 21*n + 6)*x)*v[i - 1] - ((2*n - 1)*x + (6*n - 3)*x^2)*deriv(v[i - 1], x))/(n^2 - 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