Real-rooted atlas

A225117 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 3\,n\right)\,P_{n}(x) = \left(50 - 220\,x + 440\,x^{2} - 30\,n + 192\,n x - 549\,n x^{2} - 36\,n^{2} x + 216\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 150\,x^{2} + 120\,x^{3} - 18\,n x + 135\,n x^{2} - 117\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-50 + 128\,x + 21\,n - 96\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-24\,x + 24\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
313215
4931203216
5545226275343564
6293328996320344072568040
7151773082267873143559922044561487160
87710129514721467086115481408346249598256736741600

All 15 rows

Technical details
Node id
n-8eee865029988ec1 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5c9708a73b43501c

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 -2)
  • 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:       0;
   2:       1;
   3:      13,         2;
   4:      93,       120,          3;
   5:     545,      2262,        753,           4;
   6:    2933,     28996,      32034,        4072,           5;
   7:   15177,    308226,     787314,      355992,       20445,          6;
   8:   77101,   2951472,   14670861,    15481408,     3462495,      98256,         7;
   9:  388321,  26541614,  232214799,   471034636,   252380015,   31050078,    458689,       8;
  10: 1948005, 229419036, 3310798788, 11548400616, 11944773090, 3649561740, 264125484, 2097072, 9;
%F Formula
The row polynomials satisfy (3*n-8)*P(n,x) = -(30*n-50+(36*n^2-192*n+220)*x+(27*n^3-216*n^2+549*n-440)*x^2)*P(n-2,x) - ((18*n-30)*x+(27*n^2-135*n+150)*x^2-(27*n^2-117*n+120)*x^3)*d/dx P(n-2,x) + (21*n-50+(18*n^2-96*n+128)*x)*P(n-1,x) + ((9*n-24)*x-(9*n-24)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 50 + (36*n^2 - 192*n + 220)*x + (27*n^3 - 216*n^2 + 549*n - 440)*x^2)*P[n - 2, x] - ((18*n - 30)*x + (27*n^2 - 135*n + 150)*x^2 - (27*n^2 - 117*n + 120)*x^3)*D[P[n - 2, x], x] + (21*n - 50 + (18*n^2 - 96*n + 128)*x)*P[n - 1, x] + ((9*n - 24)*x - (9*n - 24)*x^2)*D[P[n - 1, x], x])/(3*n - 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(30*n - 50 + (36*n^2 - 192*n + 220)*x + (27*n^3 - 216*n^2 + 549*n - 440)*x^2)*v[i - 2] - ((18*n - 30)*x + (27*n^2 - 135*n + 150)*x^2 - (27*n^2 - 117*n + 120)*x^3)*deriv(v[i - 2], x) + (21*n - 50 + (18*n^2 - 96*n + 128)*x)*v[i - 1] + ((9*n - 24)*x - (9*n - 24)*x^2)*deriv(v[i - 1], x))/(3*n - 8)); 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. [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.