Real-rooted atlas

A306364 → D+1

nonnegative coefficients

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

Recurrence

\(\left(10 - 14\,x + 4\,x^{2} - 3\,n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(16 - 32\,x + 18\,x^{2} - 4\,x^{3} + 6\,n + n x - 10\,n x^{2} + 4\,n x^{3} + n^{2} + 4\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 20\,x^{2} + 20\,x^{3} - 8\,x^{4} - 2\,n x + 10\,n x^{2} - 8\,n x^{3} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
133
215318
310545150
4900630451575
588209450157519845
695760151200425251575291060
7113400025401601091475992254864860
814515200444528002778300043659009922591216125
919958400080831520070727580017027010098232751895268375
10293388480015268176000180748260006286896000638512875982327543212118950
11459242784002996753760004655671020002247565320003511820812514047283251072446224850
1276281004800061133776704001213685272800078664786200001769957689500126425549250140472832528777307033475
1313403662272000129623221728000321401840760000271730647188000843679831995009313348794750273922023375830114625965625
1424845812992000028559608197120008672006030688000932649705187200038655875938680006135853323600003259672078162527392202337525614965601225000
Technical details
Node id
n-679ddd982dbce61c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-eac7d4e10a398dd8

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:        15,         3;
  3:       105,        45;
  4:       900,       630,        45;
  5:      8820,      9450,      1575;
  6:     95760,    151200,     42525,      1575;
  7:   1134000,   2540160,   1091475,     99225;
  8:  14515200,  44452800,  27783000,   4365900,   99225;
  9: 199584000, 808315200, 707275800, 170270100, 9823275;
%F Formula
The row polynomials satisfy (n^2-3*n+10+(4*n-14)*x+4*x^2)*P(n,x) = (n^3+n^2+6*n+16+(n^3+n-32)*x+(4*n^2-10*n+18)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((2*n^2-2*n+8)*x-(2*n^2-10*n+20)*x^2-(8*n-20)*x^3-8*x^4)*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[((n^3 + n^2 + 6*n + 16 + (n^3 + n - 32)*x + (4*n^2 - 10*n + 18)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((2*n^2 - 2*n + 8)*x - (2*n^2 - 10*n + 20)*x^2 - (8*n - 20)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^2 - 3*n + 10 + (4*n - 14)*x + 4*x^2)]];
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] = ((n^3 + n^2 + 6*n + 16 + (n^3 + n - 32)*x + (4*n^2 - 10*n + 18)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((2*n^2 - 2*n + 8)*x - (2*n^2 - 10*n + 20)*x^2 - (8*n - 20)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^2 - 3*n + 10 + (4*n - 14)*x + 4*x^2)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.