Real-rooted atlas

A172431 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,x + 2\,n - 12\,n x - 6\,n^{2} + 26\,n^{2} x + 4\,n^{3} - 24\,n^{3} x + 8\,n^{4} x\right)\,P_{n}(x) = \left(8\,x^{2} - 6\,n x + 6\,n x^{2} - 2\,n^{2} - 18\,n^{2} x - 10\,n^{2} x^{2} + 4\,n^{3} + 8\,n^{3} x - 32\,n^{3} x^{2} + 8\,n^{4} x + 24\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(x^{2} + 4\,x^{3} + n x^{2} + 4\,n x^{3} - 2\,n^{2} x - 8\,n^{2} x^{2} - 4\,n^{3} x^{2} - 16\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
241216
36403682
488418080352
5101445045601501368
6122201080201614002524980
714312198052806300302439217302
8164203276114401980016632588057658040
91854450402184050050617763880810560810189446
102068473443808010920018018016816882368178201100604964
112284010260620162142004455365605604118401621622860014521897488
122410121386095760387600976752155937615558409266403003004404418725863080
13261200182161416806583501953504379848048372483938220194480052852865520236617888138
142814042340020240010626003634092835665613023360136047609237800385070489107294640294053985856
Technical details
Node id
n-a2f64418470a1afb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f7ea65e9d1891b99

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  2;
  2:  4,  12;
  3:  6,  40,   36;
  4:  8,  84,  180,    80;
  5: 10, 144,  504,   560,   150;
  6: 12, 220, 1080,  2016,  1400,   252;
  7: 14, 312, 1980,  5280,  6300,  3024,   392;
  8: 16, 420, 3276, 11440, 19800, 16632,  5880,   576;
  9: 18, 544, 5040, 21840, 50050, 61776, 38808, 10560, 810;
%F Formula
The row polynomials satisfy (4*n^3-6*n^2+2*n+(8*n^4-24*n^3+26*n^2-12*n+2)*x)*P(n,x) = (4*n^3-2*n^2+(8*n^4+8*n^3-18*n^2-6*n)*x+(24*n^4-32*n^3-10*n^2+6*n+8)*x^2)*P(n-1,x) - (2*n^2*x+(4*n^3+8*n^2-n-1)*x^2+(16*n^3-4*n-4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 2*n^2 + (8*n^4 + 8*n^3 - 18*n^2 - 6*n)*x + (24*n^4 - 32*n^3 - 10*n^2 + 6*n + 8)*x^2)*P[n - 1, x] - (2*n^2*x + (4*n^3 + 8*n^2 - n - 1)*x^2 + (16*n^3 - 4*n - 4)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 + 2*n + (8*n^4 - 24*n^3 + 26*n^2 - 12*n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^3 - 2*n^2 + (8*n^4 + 8*n^3 - 18*n^2 - 6*n)*x + (24*n^4 - 32*n^3 - 10*n^2 + 6*n + 8)*x^2)*v[i - 1] - (2*n^2*x + (4*n^3 + 8*n^2 - n - 1)*x^2 + (16*n^3 - 4*n - 4)*x^3)*deriv(v[i - 1], x))/(4*n^3 - 6*n^2 + 2*n + (8*n^4 - 24*n^3 + 26*n^2 - 12*n + 2)*x)); 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