Real-rooted atlas

A086645 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12 - 6\,x - 14\,n + 13\,n x + 4\,n^{2} - 9\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(48 + 48\,x + 4\,x^{2} - 56\,n - 32\,n x + 12\,n x^{2} + 16\,n^{2} - 12\,n^{2} x - 19\,n^{2} x^{2} + 8\,n^{3} x + 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(12\,x + 14\,x^{2} + 2\,x^{3} - 8\,n x - 4\,n x^{2} + 4\,n x^{3} - 4\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3816327
43296544186
512851248012851253
651225603360160025068288
720481228820736143364200432754047
88192573441182721075204704094086868348470

All 15 rows

Technical details
Node id
n-3bbd2c9c2d5cb52d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4bc79956b2872ab3

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:      2,       2;
   3:      8,      16,       3;
   4:     32,      96,      54,       4;
   5:    128,     512,     480,     128,       5;
   6:    512,    2560,    3360,    1600,     250,       6;
   7:   2048,   12288,   20736,   14336,    4200,     432,      7;
   8:   8192,   57344,  118272,  107520,   47040,    9408,    686,     8;
   9:  32768,  262144,  638976,  720896,  422400,  129024,  18816,  1024,    9;
  10: 131072, 1179648, 3317760, 4472832, 3294720, 1368576, 310464, 34560, 1458, 10;
%F Formula
The row polynomials satisfy (4*n^2-14*n+12+(2*n^3-9*n^2+13*n-6)*x)*P(n,x) = (16*n^2-56*n+48+(8*n^3-12*n^2-32*n+48)*x+(6*n^3-19*n^2+12*n+4)*x^2)*P(n-1,x) - ((8*n-12)*x+(4*n^2+4*n-14)*x^2+(4*n^2-4*n-2)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 56*n + 48 + (8*n^3 - 12*n^2 - 32*n + 48)*x + (6*n^3 - 19*n^2 + 12*n + 4)*x^2)*P[n - 1, x] - ((8*n - 12)*x + (4*n^2 + 4*n - 14)*x^2 + (4*n^2 - 4*n - 2)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (2*n^3 - 9*n^2 + 13*n - 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = ((16*n^2 - 56*n + 48 + (8*n^3 - 12*n^2 - 32*n + 48)*x + (6*n^3 - 19*n^2 + 12*n + 4)*x^2)*v[i - 1] - ((8*n - 12)*x + (4*n^2 + 4*n - 14)*x^2 + (4*n^2 - 4*n - 2)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (2*n^3 - 9*n^2 + 13*n - 6)*x)); 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