Real-rooted atlas

A111125 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12 + 12\,x - 14\,n - 32\,n x + 4\,n^{2} + 31\,n^{2} x - 13\,n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(-2\,n - 6\,n x - n x^{2} + 4\,n^{2} + 11\,n^{2} x + 3\,n^{2} x^{2} - 7\,n^{3} x - 3\,n^{3} x^{2} + 2\,n^{4} x + n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 6\,x^{2} - x^{3} + 8\,n x + 14\,n x^{2} + 3\,n x^{3} - 12\,n^{2} x^{2} - 3\,n^{2} x^{3} + 4\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
3549
41428951
5301088116235
65530839617625960
79172814041040325363624
814015124050440022505404912941
9204285610098149601105042848336444349
102855016225724347243225239407448121681147255
1138583164633211211214332510584046648120961701100476855
125061315688803263120418600394128229908839041863023001211513176
1365020020160875572000110500012852009496204560001417502750030251444721784
14819294842779921166880268515037674723418632206841684199522770039204388816914527801
1510154222446144822559686088550101163601101559281459844160079146740035090054288490119644164905
Technical details
Node id
n-679d41c81ca02564 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6c6f5aa637e12da4

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 -2)
  • 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:   0;
   2:   1;
   3:   5,    4;
   4:  14,   28,     9;
   5:  30,  108,    81,    16;
   6:  55,  308,   396,   176,    25;
   7:  91,  728,  1404,  1040,   325,    36;
   8: 140, 1512,  4050,  4400,  2250,   540,   49;
   9: 204, 2856, 10098, 14960, 11050,  4284,  833,   64;
  10: 285, 5016, 22572, 43472, 43225, 23940, 7448, 1216, 81;
%F Formula
The row polynomials satisfy (4*n^2-14*n+12+(2*n^4-13*n^3+31*n^2-32*n+12)*x)*P(n,x) = (4*n^2-2*n+(2*n^4-7*n^3+11*n^2-6*n)*x+(n^4-3*n^3+3*n^2-n)*x^2)*P(n-1,x) + ((8*n-8)*x+(4*n^3-12*n^2+14*n-6)*x^2+(n^3-3*n^2+3*n-1)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n + (2*n^4 - 7*n^3 + 11*n^2 - 6*n)*x + (n^4 - 3*n^3 + 3*n^2 - n)*x^2)*P[n - 1, x] + ((8*n - 8)*x + (4*n^3 - 12*n^2 + 14*n - 6)*x^2 + (n^3 - 3*n^2 + 3*n - 1)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (2*n^4 - 13*n^3 + 31*n^2 - 32*n + 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 2*n + (2*n^4 - 7*n^3 + 11*n^2 - 6*n)*x + (n^4 - 3*n^3 + 3*n^2 - n)*x^2)*v[i - 1] + ((8*n - 8)*x + (4*n^3 - 12*n^2 + 14*n - 6)*x^2 + (n^3 - 3*n^2 + 3*n - 1)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (2*n^4 - 13*n^3 + 31*n^2 - 32*n + 12)*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