Real-rooted atlas

A086645 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-40 + 62\,x - 12\,x^{2} + 66\,n - 55\,n x + 6\,n x^{2} - 35\,n^{2} + 12\,n^{2} x + 6\,n^{3}\right)\,P_{n}(x) = \left(-40 + 86\,x + 402\,x^{2} - 108\,x^{3} + 156\,n + 203\,n x - 417\,n x^{2} + 54\,n x^{3} - 116\,n^{2} - 231\,n^{2} x + 108\,n^{2} x^{2} + 24\,n^{3} + 54\,n^{3} x\right)\,P_{n-1}(x) + \left(-8\,x + 14\,x^{2} + 102\,x^{3} - 36\,x^{4} + 28\,n x + 60\,n x^{2} - 72\,n x^{3} - 12\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2538
33242983
417646824327914
5896364840321188819845
643522400042480270005265243103340
720480142848352512368928155520218707291062887
89420879564825320963773952266716880967686751218710761686

All 15 rows

Technical details
Node id
n-725d541f72da879f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4dd85c739e1acd3d

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:       5,        3;
   3:      32,       42,         9;
   4:     176,      468,       243,        27;
   5:     896,     3648,      4032,      1188,        81;
   6:    4352,    24000,     42480,     27000,      5265,       243;
   7:   20480,   142848,    352512,    368928,    155520,     21870,      729;
   8:   94208,   795648,   2532096,   3773952,   2667168,    809676,    86751,     2187;
   9:  425984,  4227072,  16515072,  32237568,  32970240,  16982784,  3919104,   332424,    6561;
  10: 1900544, 21676032, 100528128, 243689472, 330511104, 249422976, 98467488, 17950896, 1240029, 19683;
%F Formula
The row polynomials satisfy (6*n^3-35*n^2+66*n-40+(12*n^2-55*n+62)*x+(6*n-12)*x^2)*P(n,x) = (24*n^3-116*n^2+156*n-40+(54*n^3-231*n^2+203*n+86)*x+(108*n^2-417*n+402)*x^2+(54*n-108)*x^3)*P(n-1,x) - ((12*n^2-28*n+8)*x+(36*n^2-60*n-14)*x^2+(72*n-102)*x^3+36*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((24*n^3 - 116*n^2 + 156*n - 40 + (54*n^3 - 231*n^2 + 203*n + 86)*x + (108*n^2 - 417*n + 402)*x^2 + (54*n - 108)*x^3)*P[n - 1, x] - ((12*n^2 - 28*n + 8)*x + (36*n^2 - 60*n - 14)*x^2 + (72*n - 102)*x^3 + 36*x^4)*D[P[n - 1, x], x])/(6*n^3 - 35*n^2 + 66*n - 40 + (12*n^2 - 55*n + 62)*x + (6*n - 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = ((24*n^3 - 116*n^2 + 156*n - 40 + (54*n^3 - 231*n^2 + 203*n + 86)*x + (108*n^2 - 417*n + 402)*x^2 + (54*n - 108)*x^3)*v[i - 1] - ((12*n^2 - 28*n + 8)*x + (36*n^2 - 60*n - 14)*x^2 + (72*n - 102)*x^3 + 36*x^4)*deriv(v[i - 1], x))/(6*n^3 - 35*n^2 + 66*n - 40 + (12*n^2 - 55*n + 62)*x + (6*n - 12)*x^2)); 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. [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