Real-rooted atlas

narayana → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4 - 11\,x + 3\,x^{2} - 3\,n - 5\,n x + 3\,n x^{2} - 4\,n^{2} + 6\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(-2 + 13\,x + 66\,x^{2} - 27\,x^{3} + 12\,n - 2\,n x - 114\,n x^{2} + 27\,n x^{3} - 22\,n^{2} - 48\,n^{2} x + 54\,n^{2} x^{2} + 12\,n^{3} + 27\,n^{3} x\right)\,P_{n-1}(x) + \left(-x + 7\,x^{2} + 24\,x^{3} - 18\,x^{4} + 5\,n x + 3\,n x^{2} - 36\,n x^{3} - 6\,n^{2} x - 18\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2538
32033962
47722516227491
529412601638702813975
611226390124209720283524332730
742903059180190951755103010935729272940
8164451411414684687588356237002466454082421872298245

All 15 rows

Technical details
Node id
n-d4ba698c2649182f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f603151ffaefdb02

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:      5,       3;
   3:     20,      33,        9;
   4:     77,     225,      162,       27;
   5:    294,    1260,     1638,      702,       81;
   6:   1122,    6390,    12420,     9720,     2835,      243;
   7:   4290,   30591,    80190,    95175,    51030,    10935,      729;
   8:  16445,  141141,   468468,   758835,   623700,   246645,    40824,    2187;
   9:  63206,  634920,  2558556,  5297292,  5999994,  3667356,  1122660,  148716,   6561;
  10: 243542, 2804490, 13318344, 33729696, 49223538, 41793570, 19901700, 4881384, 531441, 19683;
%F Formula
The row polynomials satisfy (3*n^3-4*n^2-3*n+4+(6*n^2-5*n-11)*x+(3*n+3)*x^2)*P(n,x) = (12*n^3-22*n^2+12*n-2+(27*n^3-48*n^2-2*n+13)*x+(54*n^2-114*n+66)*x^2+(27*n-27)*x^3)*P(n-1,x) - ((6*n^2-5*n+1)*x+(18*n^2-3*n-7)*x^2+(36*n-24)*x^3+18*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[((12*n^3 - 22*n^2 + 12*n - 2 + (27*n^3 - 48*n^2 - 2*n + 13)*x + (54*n^2 - 114*n + 66)*x^2 + (27*n - 27)*x^3)*P[n - 1, x] - ((6*n^2 - 5*n + 1)*x + (18*n^2 - 3*n - 7)*x^2 + (36*n - 24)*x^3 + 18*x^4)*D[P[n - 1, x], x])/(3*n^3 - 4*n^2 - 3*n + 4 + (6*n^2 - 5*n - 11)*x + (3*n + 3)*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] = ((12*n^3 - 22*n^2 + 12*n - 2 + (27*n^3 - 48*n^2 - 2*n + 13)*x + (54*n^2 - 114*n + 66)*x^2 + (27*n - 27)*x^3)*v[i - 1] - ((6*n^2 - 5*n + 1)*x + (18*n^2 - 3*n - 7)*x^2 + (36*n - 24)*x^3 + 18*x^4)*deriv(v[i - 1], x))/(3*n^3 - 4*n^2 - 3*n + 4 + (6*n^2 - 5*n - 11)*x + (3*n + 3)*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