Real-rooted atlas

narayana → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(1 + 2\,x + n + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
315511
419211445
5114568442197
6120120300330132903
7127225825148512874294279
81353851925500570075005143020793

All 15 rows

Technical details
Node id
n-a77827aa20dca9a0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-1e5fbd8fe1b88630

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: 1,  2;
   3: 1,  5,   5;
   4: 1,  9,  21,   14;
   5: 1, 14,  56,   84,    42;
   6: 1, 20, 120,  300,   330,   132;
   7: 1, 27, 225,  825,  1485,  1287,    429;
   8: 1, 35, 385, 1925,  5005,  7007,   5005,   1430;
   9: 1, 44, 616, 4004, 14014, 28028,  32032,  19448,  4862;
  10: 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (n+1+(2*n+2)*x)*P(n-1,x) + (2*x+2*x^2)*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[((n + 1 + (2*n + 2)*x)*P[n - 1, x] + (2*x + 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];
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] = ((n + 1 + (2*n + 2)*x)*v[i - 1] + (2*x + 2*x^2)*deriv(v[i - 1], x))/(n + 1)); 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

Candidate equalities ?

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