Real-rooted atlas

A085478 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(90 + 216\,x - 162\,x^{2} - 60\,n - 144\,n x + 351\,n x^{2} - 243\,n^{2} x^{2} + 54\,n^{3} x^{2}\right)\,P_{n}(x) = \left(120 + 378\,x + 216\,x^{2} - 90\,n - 261\,n x + 54\,n x^{2} + 162\,n x^{3} - 45\,n^{2} x - 351\,n^{2} x^{2} - 243\,n^{2} x^{3} + 81\,n^{3} x^{2} + 81\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 210\,x - 477\,x^{2} - 216\,x^{3} - 90\,n x^{2} - 324\,n x^{3} - 162\,n x^{4} + 45\,n^{2} x^{2} + 162\,n^{2} x^{3} + 81\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2268
35302762
413114216108451
534390113411884053151
6891260494179925670145821410
72333924193594255246575247865103142532

All 15 rows

Technical details
Node id
n-83c07888516f09b4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-087a0ca9a03d3880

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    2,     6;
  3:    5,    30,     27;
  4:   13,   114,    216,    108;
  5:   34,   390,   1134,   1188,     405;
  6:   89,  1260,   4941,   7992,    5670,    1458;
  7:  233,  3924,  19359,  42552,   46575,   24786,    5103;
  8:  610, 11910,  70794, 197100,  293625,  240570,  102060,  17496;
  9: 1597, 35466, 246618, 831924, 1573425, 1753974, 1143072, 402408, 59049;
%F Formula
The row polynomials satisfy -(60*n-90+(144*n-216)*x-(54*n^3-243*n^2+351*n-162)*x^2)*P(n,x) = -(90*n-120+(45*n^2+261*n-378)*x-(81*n^3-351*n^2+54*n+216)*x^2-(81*n^3-243*n^2+162*n)*x^3)*P(n-1,x) - (25+210*x-(45*n^2-90*n-477)*x^2-(162*n^2-324*n-216)*x^3-(81*n^2-162*n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(90*n - 120 + (45*n^2 + 261*n - 378)*x - (81*n^3 - 351*n^2 + 54*n + 216)*x^2 - (81*n^3 - 243*n^2 + 162*n)*x^3)*P[n - 1, x] - (25 + 210*x - (45*n^2 - 90*n - 477)*x^2 - (162*n^2 - 324*n - 216)*x^3 - (81*n^2 - 162*n)*x^4)*D[P[n - 1, x], x])/(-60*n + 90 - (144*n - 216)*x + (54*n^3 - 243*n^2 + 351*n - 162)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(90*n - 120 + (45*n^2 + 261*n - 378)*x - (81*n^3 - 351*n^2 + 54*n + 216)*x^2 - (81*n^3 - 243*n^2 + 162*n)*x^3)*v[i - 1] - (25 + 210*x - (45*n^2 - 90*n - 477)*x^2 - (162*n^2 - 324*n - 216)*x^3 - (81*n^2 - 162*n)*x^4)*deriv(v[i - 1], x))/(-60*n + 90 - (144*n - 216)*x + (54*n^3 - 243*n^2 + 351*n - 162)*x^2)); 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

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