Real-rooted atlas

A081577 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,x^{2} + 2\,n - 3\,n x - 3\,n x^{2} + 3\,n^{2} x\right)\,P_{n}(x) = \left(-16 - 66\,x - 66\,x^{2} - 18\,x^{3} + 26\,n + 87\,n x + 42\,n x^{2} - 6\,n^{2} - 15\,n^{2} x\right)\,P_{n-1}(x) + \left(-4 - 28\,x - 63\,x^{2} - 48\,x^{3} - 9\,x^{4} + 4\,n + 24\,n x + 39\,n x^{2} + 9\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2181836
37218081333
4264104411343242766
5900489687485832121521591
62952204485119256376267304374162072
7940879200255960399168311040113724153091183809
8293762911681154736234446425952401548396459270524888475138
990288102988848444481215388817598060149065927164612178459217714759749515
102740803536352192533765761238410354716011494872078259608313878246731586590490416141580
1182368011860416733782242553016325494910407591864326775151043837922561317973682480058019486172869895349
1224549123902457627046224010733679362695550400447197760049733429763689906400178422458453498394089636382637729219631309238
137265856126383616970035264432621388812429974880241226449923226794278429844676800188760757327945633440211230082831886460020726199133368738879
14213776644038704643400942464168430026245452461216012134511763219021567219221180503635216695325303291776317760341553587768150179176111921474666961566900780916608
Technical details
Node id
n-3ac142214cb04c13 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6873db4a93d27ca5

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:    18,      18;
  3:    72,     180,      81;
  4:   264,    1044,    1134,      324;
  5:   900,    4896,    8748,     5832,     1215;
  6:  2952,   20448,   51192,    56376,    26730,     4374;
  7:  9408,   79200,  255960,   399168,   311040,   113724,   15309;
  8: 29376,  291168, 1154736,  2344464,  2595240,  1548396,  459270,   52488;
  9: 90288, 1029888, 4844448, 12153888, 17598060, 14906592, 7164612, 1784592, 177147;
%F Formula
The row polynomials satisfy (2*n-2+(3*n^2-3*n)*x-(3*n-3)*x^2)*P(n,x) = -(6*n^2-26*n+16+(15*n^2-87*n+66)*x-(42*n-66)*x^2+18*x^3)*P(n-1,x) + (4*n-4+(24*n-28)*x+(39*n-63)*x^2+(9*n-48)*x^3-9*x^4)*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[(-(6*n^2 - 26*n + 16 + (15*n^2 - 87*n + 66)*x - (42*n - 66)*x^2 + 18*x^3)*P[n - 1, x] + (4*n - 4 + (24*n - 28)*x + (39*n - 63)*x^2 + (9*n - 48)*x^3 - 9*x^4)*D[P[n - 1, x], x])/(2*n - 2 + (3*n^2 - 3*n)*x - (3*n - 3)*x^2)]];
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] = (-(6*n^2 - 26*n + 16 + (15*n^2 - 87*n + 66)*x - (42*n - 66)*x^2 + 18*x^3)*v[i - 1] + (4*n - 4 + (24*n - 28)*x + (39*n - 63)*x^2 + (9*n - 48)*x^3 - 9*x^4)*deriv(v[i - 1], x))/(2*n - 2 + (3*n^2 - 3*n)*x - (3*n - 3)*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.