Real-rooted atlas

A207607 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(23 + 44\,x + 25\,x^{2} + 4\,x^{3} - 7\,n - 24\,n x - 22\,n x^{2} - 4\,n x^{3} - 15\,n^{2} - 12\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(77 + 159\,x + 103\,x^{2} + 25\,x^{3} + 2\,x^{4} - 44\,n - 84\,n x - 64\,n x^{2} - 19\,n x^{3} - 2\,n x^{4} - 21\,n^{2} - 16\,n^{2} x - 2\,n^{2} x^{2} + 3\,n^{3} + 7\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-100 - 280\,x - 287\,x^{2} - 132\,x^{3} - 27\,x^{4} - 2\,x^{5} - 20\,n - 34\,n x - 16\,n x^{2} - 2\,n x^{3} + 5\,n^{2} + 16\,n^{2} x + 13\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
21713232
35463212140
4166263139292599
549910037352453722521
6147536003393157838145210474
7430312370142958596290054753243066
812421411295638941761182504809743612175565
935544133275211647186657100656343717413969692710603
10100976423019764075782990503756212415593531082012257722858708
11285095132002326734743124454233983311781354065469596815138151185211440264
128006774060872911701811974788102435756009114247689472192614736620475182793245574627
132238329123425543042924644392336427405252866652813772478479494212080972170752693921731012180831325
14623212137125637997242921600375421713669011294797877116002428881024869052919265623090013463825491092714970718
Technical details
Node id
n-2a28111c2ddad160 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-246c2586d8139640

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:     5,      2;
  2:    17,     13,      2;
  3:    54,     63,     21,      2;
  4:   166,    263,    139,     29,      2;
  5:   499,   1003,    735,    245,     37,     2;
  6:  1475,   3600,   3393,   1578,    381,    45,    2;
  7:  4303,  12370,  14295,   8596,   2900,   547,   53,   2;
  8: 12421,  41129,  56389,  41761,  18250,  4809,  743,  61,  2;
  9: 35544, 133275, 211647, 186657, 100656, 34371, 7413, 969, 69, 2;
%F Formula
The row polynomials satisfy (2*n^3-15*n^2-7*n+23+(4*n^3-12*n^2-24*n+44)*x-(22*n-25)*x^2-(4*n-4)*x^3)*P(n,x) = (3*n^3-21*n^2-44*n+77+(7*n^3-16*n^2-84*n+159)*x+(2*n^3-2*n^2-64*n+103)*x^2-(19*n-25)*x^3-(2*n-2)*x^4)*P(n-1,x) + (5*n^2-20*n-100+(16*n^2-34*n-280)*x+(13*n^2-16*n-287)*x^2+(2*n^2-2*n-132)*x^3-27*x^4-2*x^5)*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[((3*n^3 - 21*n^2 - 44*n + 77 + (7*n^3 - 16*n^2 - 84*n + 159)*x + (2*n^3 - 2*n^2 - 64*n + 103)*x^2 - (19*n - 25)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (5*n^2 - 20*n - 100 + (16*n^2 - 34*n - 280)*x + (13*n^2 - 16*n - 287)*x^2 + (2*n^2 - 2*n - 132)*x^3 - 27*x^4 - 2*x^5)*D[P[n - 1, x], x])/(2*n^3 - 15*n^2 - 7*n + 23 + (4*n^3 - 12*n^2 - 24*n + 44)*x - (22*n - 25)*x^2 - (4*n - 4)*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] = ((3*n^3 - 21*n^2 - 44*n + 77 + (7*n^3 - 16*n^2 - 84*n + 159)*x + (2*n^3 - 2*n^2 - 64*n + 103)*x^2 - (19*n - 25)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (5*n^2 - 20*n - 100 + (16*n^2 - 34*n - 280)*x + (13*n^2 - 16*n - 287)*x^2 + (2*n^2 - 2*n - 132)*x^3 - 27*x^4 - 2*x^5)*deriv(v[i - 1], x))/(2*n^3 - 15*n^2 - 7*n + 23 + (4*n^3 - 12*n^2 - 24*n + 44)*x - (22*n - 25)*x^2 - (4*n - 4)*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