Real-rooted atlas

A060920 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\,x + 11\,x^{2} - 5\,x^{3} + n - 5\,n x - 32\,n x^{2} + 10\,n x^{3} - n^{2} - 19\,n^{2} x + 20\,n^{2} x^{2} - 2\,n^{3} + 10\,n^{3} x\right)\,P_{n}(x) = \left(2 + 14\,x + 10\,x^{2} - 13\,x^{3} + 10\,x^{4} + 25\,x^{5} - n - 26\,n x - 56\,n x^{2} + 26\,n x^{3} + 50\,n x^{4} - 25\,n x^{5} - 5\,n^{2} - 27\,n^{2} x + 27\,n^{2} x^{2} + 25\,n^{2} x^{3} - 50\,n^{2} x^{4} - 2\,n^{3} + 11\,n^{3} x - 25\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 17\,x^{2} + 39\,x^{3} + 66\,x^{4} + 65\,x^{5} + 25\,x^{6} + 12\,n x^{2} + 67\,n x^{3} + 105\,n x^{4} + 50\,n x^{5} - 4\,n^{2} x + 11\,n^{2} x^{2} + 40\,n^{2} x^{3} + 25\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2615526
310495913131
41511626420734636
5212308041173680892997
628407197243624662214223313806
7366654200127352047017232655561062503

All 15 rows

Technical details
Node id
n-cc3f87147a27714f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-edff8dc3393a5714

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:  3,    2;
  2:  6,   15,     5;
  3: 10,   49,    59,    13;
  4: 15,  116,   264,   207,     34;
  5: 21,  230,   804,  1173,    680,     89;
  6: 28,  407,  1972,  4362,   4662,   2142,    233;
  7: 36,  665,  4200, 12735,  20470,  17232,   6555,    610;
  8: 45, 1024,  8092, 31674,  69295,  87240,  60528,  19641,  1597;
  9: 55, 1506, 14460, 70206, 197017, 335900, 347092, 204719, 57913, 4181;
%F Formula
The row polynomials satisfy -(2*n^3+n^2-n-(10*n^3-19*n^2-5*n+6)*x-(20*n^2-32*n+11)*x^2-(10*n-5)*x^3)*P(n,x) = -(2*n^3+5*n^2+n-2-(11*n^3-27*n^2-26*n+14)*x-(27*n^2-56*n+10)*x^2+(25*n^3-25*n^2-26*n+13)*x^3+(50*n^2-50*n-10)*x^4+(25*n-25)*x^5)*P(n-1,x) - ((4*n^2-4)*x-(11*n^2+12*n+17)*x^2-(40*n^2+67*n+39)*x^3-(25*n^2+105*n+66)*x^4-(50*n+65)*x^5-25*x^6)*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[(-(2*n^3 + 5*n^2 + n - 2 - (11*n^3 - 27*n^2 - 26*n + 14)*x - (27*n^2 - 56*n + 10)*x^2 + (25*n^3 - 25*n^2 - 26*n + 13)*x^3 + (50*n^2 - 50*n - 10)*x^4 + (25*n - 25)*x^5)*P[n - 1, x] - ((4*n^2 - 4)*x - (11*n^2 + 12*n + 17)*x^2 - (40*n^2 + 67*n + 39)*x^3 - (25*n^2 + 105*n + 66)*x^4 - (50*n + 65)*x^5 - 25*x^6)*D[P[n - 1, x], x])/(-2*n^3 - n^2 + n + (10*n^3 - 19*n^2 - 5*n + 6)*x + (20*n^2 - 32*n + 11)*x^2 + (10*n - 5)*x^3)]];
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] = (-(2*n^3 + 5*n^2 + n - 2 - (11*n^3 - 27*n^2 - 26*n + 14)*x - (27*n^2 - 56*n + 10)*x^2 + (25*n^3 - 25*n^2 - 26*n + 13)*x^3 + (50*n^2 - 50*n - 10)*x^4 + (25*n - 25)*x^5)*v[i - 1] - ((4*n^2 - 4)*x - (11*n^2 + 12*n + 17)*x^2 - (40*n^2 + 67*n + 39)*x^3 - (25*n^2 + 105*n + 66)*x^4 - (50*n + 65)*x^5 - 25*x^6)*deriv(v[i - 1], x))/(-2*n^3 - n^2 + n + (10*n^3 - 19*n^2 - 5*n + 6)*x + (20*n^2 - 32*n + 11)*x^2 + (10*n - 5)*x^3)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023