Real-rooted atlas

A114656 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - 6\,n + 18\,n x - 27\,n x^{2} + 18\,n^{2} x + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(6 + 72\,x + 162\,x^{3} - 30\,n - 72\,n x - 297\,n x^{2} - 243\,n x^{3} + 72\,n^{2} x + 162\,n^{2} x^{2} - 162\,n^{2} x^{3} + 135\,n^{3} x^{2} + 243\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2 + 9\,x^{2} + 81\,x^{4} + 12\,n x + 9\,n x^{2} - 108\,n x^{3} - 81\,n x^{4} + 18\,n^{2} x^{2} - 162\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2369
311482786
445306405108864
51971824394225924058960
690310590321303564014175145894896
74279607682385453801602612256998451031020064
82079334679416743513511620352957516635783214891749611085696

All 15 rows

Technical details
Node id
n-7426f72bbc940786 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ee30904fbedd4d92

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:      3,        6;
   3:     11,       48,       27;
   4:     45,      306,      405,       108;
   5:    197,     1824,     3942,      2592,       405;
   6:    903,    10590,    32130,     35640,     14175,      1458;
   7:   4279,    60768,   238545,    380160,    261225,     69984,      5103;
   8:  20793,   346794,  1674351,   3511620,   3529575,   1663578,    321489,    17496;
   9: 103049,  1974144, 11327148,  29598912,  39695670,  27760320,   9573228,  1399680,   59049;
  10: 518859, 11226870, 74683620, 234284400, 395737650, 373478364, 193322052, 51018336, 5845851, 196830;
%F Formula
The row polynomials satisfy -(6*n+6-(18*n^2+18*n)*x-(27*n^3-27*n)*x^2)*P(n,x) = -(30*n-6-(72*n^2-72*n+72)*x-(135*n^3+162*n^2-297*n)*x^2-(243*n^3-162*n^2-243*n+162)*x^3)*P(n-1,x) - (2-12*n*x-(18*n^2+9*n+9)*x^2+108*n*x^3+(162*n^2+81*n-81)*x^4)*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[(-(30*n - 6 - (72*n^2 - 72*n + 72)*x - (135*n^3 + 162*n^2 - 297*n)*x^2 - (243*n^3 - 162*n^2 - 243*n + 162)*x^3)*P[n - 1, x] - (2 - 12*n*x - (18*n^2 + 9*n + 9)*x^2 + 108*n*x^3 + (162*n^2 + 81*n - 81)*x^4)*D[P[n - 1, x], x])/(-6*n - 6 + (18*n^2 + 18*n)*x + (27*n^3 - 27*n)*x^2)]];
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] = (-(30*n - 6 - (72*n^2 - 72*n + 72)*x - (135*n^3 + 162*n^2 - 297*n)*x^2 - (243*n^3 - 162*n^2 - 243*n + 162)*x^3)*v[i - 1] - (2 - 12*n*x - (18*n^2 + 9*n + 9)*x^2 + 108*n*x^3 + (162*n^2 + 81*n - 81)*x^4)*deriv(v[i - 1], x))/(-6*n - 6 + (18*n^2 + 18*n)*x + (27*n^3 - 27*n)*x^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

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