Real-rooted atlas

A063007 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(1 - n\right)\,P_{n}(x) = \left(-2 + n\right)\,P_{n-2}(x) + \left(15 + 12\,x - 10\,n - 8\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2549
3376024121
43057446001601809
526418600104405600112028401
62352595820155400125440504008064458649
72134451043700211848022848001380960443520591367544041
8196182511194960272916003684576029752800143700483843840439296125700129

All 15 rows

Technical details
Node id
n-2c425b900cc995ac · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-08ad261863ddeed5

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,          4;
   3:        37,         60,         24;
   4:       305,        744,        600,        160;
   5:      2641,       8600,      10440,       5600,       1120;
   6:     23525,      95820,     155400,     125440,      50400,       8064;
   7:    213445,    1043700,    2118480,    2284800,    1380960,     443520,      59136;
   8:   1961825,   11194960,   27291600,   36845760,   29752800,   14370048,    3843840,     439296;
   9:  18205345,  118753200,  337967280,  548116800,  554067360,  357477120,  143759616,   32947200,   3294720;
  10: 170195525, 1249158420, 4064900400, 7697437440, 9347738400, 7549686144, 4055251200, 1396961280, 280051200, 24893440;
%F Formula
The row polynomials satisfy -(n-1)*P(n,x) = (n-2)*P(n-2,x) - (10*n-15+(8*n-12)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*P[n - 2, x] - (10*n - 15 + (8*n - 12)*x)*P[n - 1, 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; v[2] = 5 + 4*x; for(n = 3, N, my(i = n); v[i] = ((n - 2)*v[i - 2] - (10*n - 15 + (8*n - 12)*x)*v[i - 1])/(-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

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