Real-rooted atlas

A167884 → D

nonnegative coefficients

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

Recurrence

\(\left(-15 + 8\,n\right)\,P_{n}(x) = \left(63 - 322\,x + 2415\,x^{2} - 72\,n + 480\,n x - 4888\,n x^{2} - 128\,n^{2} x + 2880\,n^{2} x^{2} - 512\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(56\,x - 896\,x^{2} + 840\,x^{3} - 64\,n x + 1472\,n x^{2} - 1408\,n x^{3} - 512\,n^{2} x^{2} + 512\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-142 + 450\,x + 80\,n - 480\,n x + 128\,n^{2} x\right)\,P_{n-1}(x) + \left(-120\,x + 120\,x^{2} + 64\,n x - 64\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
218220
31793583540
41636121724908418720
514757288724433086590285795600
613285458822702165430011764540664270640098240
711957351108926188325298651110039820277231545717441072339064000

All 15 rows

Technical details
Node id
n-b82fcd790d61f9f2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-82041d63f3ba65ee

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 -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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        1;
  2:       18,           2;
  3:      179,         358,            3;
  4:     1636,       12172,         4908,             4;
  5:    14757,      288724,       433086,         59028,             5;
  6:   132854,     5882270,     21654300,      11764540,        664270,             6;
  7:  1195735,   110892618,    832529865,    1110039820,     277231545,       7174410,            7;
  8: 10761672,  2002356536,  27633142632,   73262628120,   46055237720,    6007069608,     75331704,         8;
  9: 96855113, 35266891720, 838001771196, 3928278525176, 4910348156470, 1676003542392, 123434121020, 774840904, 9;
%F Formula
The row polynomials satisfy (8*n-15)*P(n,x) = -(72*n-63+(128*n^2-480*n+322)*x+(512*n^3-2880*n^2+4888*n-2415)*x^2)*P(n-2,x) - ((64*n-56)*x+(512*n^2-1472*n+896)*x^2-(512*n^2-1408*n+840)*x^3)*d/dx P(n-2,x) + (80*n-142+(128*n^2-480*n+450)*x)*P(n-1,x) + ((64*n-120)*x-(64*n-120)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(72*n - 63 + (128*n^2 - 480*n + 322)*x + (512*n^3 - 2880*n^2 + 4888*n - 2415)*x^2)*P[n - 2, x] - ((64*n - 56)*x + (512*n^2 - 1472*n + 896)*x^2 - (512*n^2 - 1408*n + 840)*x^3)*D[P[n - 2, x], x] + (80*n - 142 + (128*n^2 - 480*n + 450)*x)*P[n - 1, x] + ((64*n - 120)*x - (64*n - 120)*x^2)*D[P[n - 1, x], x])/(8*n - 15)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(72*n - 63 + (128*n^2 - 480*n + 322)*x + (512*n^3 - 2880*n^2 + 4888*n - 2415)*x^2)*v[i - 2] - ((64*n - 56)*x + (512*n^2 - 1472*n + 896)*x^2 - (512*n^2 - 1408*n + 840)*x^3)*deriv(v[i - 2], x) + (80*n - 142 + (128*n^2 - 480*n + 450)*x)*v[i - 1] + ((64*n - 120)*x - (64*n - 120)*x^2)*deriv(v[i - 1], x))/(8*n - 15)); 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.