Real-rooted atlas

A391403 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-3 - 12\,x + 3\,n + 12\,n x\right)\,P_{n-2}(x) + \left(-1 + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2369
371825
4197254145
551240270561
614181013505402841
739326465670378012489
8110785682268022680567060705
93139274328618411340051030281185
10895387210317520521640357210612361353769
11256532755501140480224532020582106735966418809
12737898664484018410923076010665270538876867359630917041
132129412713464139382103660228051081030350269928756748148331665
14616227846900647729682141080940231080850201405204788107327505784716698425
Technical details
Node id
n-7213f12fdfdd9709 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-ea55a274263d0861

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:    1;
  2:    3,     6;
  3:    7,    18;
  4:   19,    72,    54;
  5:   51,   240,   270;
  6:  141,   810,  1350,    540;
  7:  393,  2646,  5670,   3780;
  8: 1107,  8568, 22680,  22680,  5670;
  9: 3139, 27432, 86184, 113400, 51030;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n-3+(12*n-12)*x)*P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (12*n - 12)*x)*P[n - 2, x] + (2*n - 1)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (12*n - 12)*x)*v[i - 2] + (2*n - 1)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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