Real-rooted atlas

A008459 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2} - 3\,n^{2} x + 6\,n^{3} x + 4\,n^{4} x^{2}\right)\,P_{n}(x) = \left(8\,x^{2} - 2\,n - 26\,n x - 88\,n x^{2} - 64\,n x^{3} + 4\,n^{2} + 24\,n^{2} x + 8\,n^{2} x^{2} - 16\,n^{2} x^{3} + 24\,n^{3} x + 112\,n^{3} x^{2} + 96\,n^{3} x^{3} + 16\,n^{4} x^{2} + 48\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} + 4\,x^{3} - 2\,n x - 20\,n x^{2} - 64\,n x^{3} - 64\,n x^{4} - 12\,n^{2} x^{2} - 72\,n^{2} x^{3} - 96\,n^{2} x^{4} - 8\,n^{3} x^{3} - 32\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121618
2696144246
320480172810243252
47022401296020480640042150
5252100808064021504019200036864534876
6924443524536001720320268800015482882007046656188
73432192192239500811827200268800002786918411239424104857681455016

All 15 rows

Technical details
Node id
n-818f755cd90b9338 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d426cb592eccfc38

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 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:     2,      16;
  2:     6,      96,      144;
  3:    20,     480,     1728,      1024;
  4:    70,    2240,    12960,     20480,       6400;
  5:   252,   10080,    80640,    215040,     192000,      36864;
  6:   924,   44352,   453600,   1720320,    2688000,    1548288,     200704;
  7:  3432,  192192,  2395008,  11827200,   26880000,   27869184,   11239424,    1048576;
  8: 12870,  823680, 12108096,  73801728,  221760000,  340623360,  252887040,   75497472,   5308416;
  9: 48620, 3500640, 59304960, 430510080, 1614412800, 3321077760, 3709009920, 2076180480, 477757440, 26214400;
%F Formula
The row polynomials satisfy (n^2+(6*n^3-3*n^2)*x+4*n^4*x^2)*P(n,x) = (4*n^2-2*n+(24*n^3+24*n^2-26*n)*x+(16*n^4+112*n^3+8*n^2-88*n+8)*x^2+(48*n^4+96*n^3-16*n^2-64*n)*x^3)*P(n-1,x) - (2*n*x+(12*n^2+20*n-1)*x^2+(8*n^3+72*n^2+64*n-4)*x^3+(32*n^3+96*n^2+64*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[((4*n^2 - 2*n + (24*n^3 + 24*n^2 - 26*n)*x + (16*n^4 + 112*n^3 + 8*n^2 - 88*n + 8)*x^2 + (48*n^4 + 96*n^3 - 16*n^2 - 64*n)*x^3)*P[n - 1, x] - (2*n*x + (12*n^2 + 20*n - 1)*x^2 + (8*n^3 + 72*n^2 + 64*n - 4)*x^3 + (32*n^3 + 96*n^2 + 64*n)*x^4)*D[P[n - 1, x], x])/(n^2 + (6*n^3 - 3*n^2)*x + 4*n^4*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] = ((4*n^2 - 2*n + (24*n^3 + 24*n^2 - 26*n)*x + (16*n^4 + 112*n^3 + 8*n^2 - 88*n + 8)*x^2 + (48*n^4 + 96*n^3 - 16*n^2 - 64*n)*x^3)*v[i - 1] - (2*n*x + (12*n^2 + 20*n - 1)*x^2 + (8*n^3 + 72*n^2 + 64*n - 4)*x^3 + (32*n^3 + 96*n^2 + 64*n)*x^4)*deriv(v[i - 1], x))/(n^2 + (6*n^3 - 3*n^2)*x + 4*n^4*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