Real-rooted atlas

A257623 → D

nonnegative coefficients

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

Recurrence

\(\left(-7 + 5\,n\right)\,P_{n}(x) = \left(48 - 144\,x + 168\,x^{2} - 120\,n + 420\,n x - 610\,n x^{2} - 150\,n^{2} x + 525\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 100\,x^{2} + 70\,x^{3} - 75\,n x + 300\,n x^{2} - 225\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-62 + 98\,x + 55\,n - 140\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-35\,x + 35\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2481866
35011002811584
44494260521348232444352
53781550046075069015126012151441440
6309324824547027024840164909401546620437453621568
72498649124513254770915835102788778031128313514991894153092252105856

All 15 rows

Technical details
Node id
n-96f444dda9cea45a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2e8e7edac39f7fae

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:         3;
  2:        48,          18;
  3:       501,        1002,           81;
  4:      4494,       26052,        13482,           324;
  5:     37815,      500460,       750690,        151260,          1215;
  6:    309324,     8245470,     27024840,      16490940,       1546620,         4374;
  7:   2498649,   124513254,    770915835,    1027887780,     311283135,     14991894,       15309;
  8:  20072298,  1783583136,  19106041698,   47282837880,   31843402830,   5350749408,   140506086,      52488;
  9: 160860507, 24712075416, 432196115796, 1800797494632, 2250996868290, 864392231592, 86492263956, 1286884056, 177147;
%F Formula
The row polynomials satisfy (5*n-7)*P(n,x) = -(120*n-48+(150*n^2-420*n+144)*x+(125*n^3-525*n^2+610*n-168)*x^2)*P(n-2,x) - ((75*n-30)*x+(125*n^2-300*n+100)*x^2-(125*n^2-225*n+70)*x^3)*d/dx P(n-2,x) + (55*n-62+(50*n^2-140*n+98)*x)*P(n-1,x) + ((25*n-35)*x-(25*n-35)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(120*n - 48 + (150*n^2 - 420*n + 144)*x + (125*n^3 - 525*n^2 + 610*n - 168)*x^2)*P[n - 2, x] - ((75*n - 30)*x + (125*n^2 - 300*n + 100)*x^2 - (125*n^2 - 225*n + 70)*x^3)*D[P[n - 2, x], x] + (55*n - 62 + (50*n^2 - 140*n + 98)*x)*P[n - 1, x] + ((25*n - 35)*x - (25*n - 35)*x^2)*D[P[n - 1, x], x])/(5*n - 7)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(120*n - 48 + (150*n^2 - 420*n + 144)*x + (125*n^3 - 525*n^2 + 610*n - 168)*x^2)*v[i - 2] - ((75*n - 30)*x + (125*n^2 - 300*n + 100)*x^2 - (125*n^2 - 225*n + 70)*x^3)*deriv(v[i - 2], x) + (55*n - 62 + (50*n^2 - 140*n + 98)*x)*v[i - 1] + ((25*n - 35)*x - (25*n - 35)*x^2)*deriv(v[i - 1], x))/(5*n - 7)); 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.