Real-rooted atlas

narayana → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 - n + n^{2}\right)\,P_{n}(x) = \left(2 + 8\,x - 6\,n - 28\,n x + 4\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} - 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
244
3203252
484288192564
53361792268810245840
6132096002304020480512059560
751484752015840023040013824024576604284
820020224224960960197120019712008601601146886122452

All 15 rows

Technical details
Node id
n-d42fbd7d701c5e4f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6c7cd585ae759b21

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:      4;
   3:     20,      32;
   4:     84,     288,      192;
   5:    336,    1792,     2688,     1024;
   6:   1320,    9600,    23040,    20480,      5120;
   7:   5148,   47520,   158400,   230400,    138240,     24576;
   8:  20020,  224224,   960960,  1971200,   1971200,    860160,    114688;
   9:  77792, 1025024,  5381376, 14350336,  20500480,  15138816,   5046272,   524288;
  10: 302328, 4582656, 28514304, 93929472, 176117760, 187858944, 107347968, 28311552, 2359296;
%F Formula
The row polynomials satisfy (n^2-n-2)*P(n,x) = (4*n^2-6*n+2+(12*n^2-28*n+8)*x)*P(n-1,x) - ((2*n-1)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (12*n^2 - 28*n + 8)*x)*P[n - 1, x] - ((2*n - 1)*x + (8*n - 4)*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] = 4; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 6*n + 2 + (12*n^2 - 28*n + 8)*x)*v[i - 1] - ((2*n - 1)*x + (8*n - 4)*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