Real-rooted atlas

A104684 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2134
3181322
41155763136
5124156384321886
61353401380250516835944
714864538001114516008898940636
816311138855370658490310084948639281488

All 15 rows

Technical details
Node id
n-7954740262dbc1d9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c880512e8efe320e

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,  3;
   3: 1,  8,   13;
   4: 1, 15,   57,    63;
   5: 1, 24,  156,   384,    321;
   6: 1, 35,  340,  1380,   2505,    1683;
   7: 1, 48,  645,  3800,  11145,   16008,    8989;
   8: 1, 63, 1113,  8855,  37065,   84903,  100849,   48639;
   9: 1, 80, 1792, 18368, 102340,  330848,  621040,  628736,  265729;
  10: 1, 99, 2736, 34944, 247716, 1055628, 2774352, 4408848, 3888657, 1462563;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n-1+(n+1)*x-(4*n-8)*x^2)*P(n-1,x) + (2*x+6*x^2+4*x^3)*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 + (n + 1)*x - (4*n - 8)*x^2)*P[n - 1, x] + (2*x + 6*x^2 + 4*x^3)*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 + (n + 1)*x - (4*n - 8)*x^2)*v[i - 1] + (2*x + 6*x^2 + 4*x^3)*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