Real-rooted atlas

A154227 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(4\,n^{2} x^{2} + 3\,n^{3} x^{2} - 3\,n^{4} x^{2} - 3\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-4}(x) + \left(-16\,n x - 12\,n x^{2} - 28\,n^{2} x - 20\,n^{2} x^{2} - 16\,n^{3} x - 12\,n^{3} x^{2} - 4\,n^{4} x - 4\,n^{4} x^{2}\right)\,P_{n-3}(x) + \left(-16 - 16\,x - 8\,x^{2} - 12\,n + 8\,n x - 4\,n x^{2} - 4\,n^{2} + 20\,n^{2} x - 4\,n^{2} x^{2} + 16\,n^{3} x + 4\,n^{4} x\right)\,P_{n-2}(x) + \left(24 + 16\,x + 16\,n + 8\,n x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 9 + 10\,x + x^{2}\)
\(P_{3}(x) = 20 + 57\,x + 22\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
2910120
32057221100
436351263391690
558124123688206214550
68733441845312124205992136160
71247657894141427454738445051301291960

All 15 rows

Technical details
Node id
n-ef8210e06ee162e1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3d22389fd257c6d1

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:   2,     1;
  2:   9,    10,      1;
  3:  20,    57,     22,       1;
  4:  36,   351,    263,      39,        1;
  5:  58,  1241,   2368,     820,       62,       1;
  6:  87,  3344,  18453,   12124,     2059,      92,      1;
  7: 124,  7657,  89414,  142745,    47384,    4505,    130,     1;
  8: 170, 15689, 324623, 1343453,   837563,  152181,   8943,   177,   1;
  9: 226, 29613, 976392, 8261662, 11911644, 3908670, 423424, 16494, 234, 1;
%F Formula
The row polynomials satisfy (4*n^2+4*n+8)*P(n,x) = -(n^6+3*n^5+3*n^4-3*n^3-4*n^2)*x^2*P(n-4,x) - ((4*n^4+16*n^3+28*n^2+16*n)*x+(4*n^4+12*n^3+20*n^2+12*n)*x^2)*P(n-3,x) - (4*n^2+12*n+16-(4*n^4+16*n^3+20*n^2+8*n-16)*x+(4*n^2+4*n+8)*x^2)*P(n-2,x) + (8*n^2+16*n+24+(8*n^2+8*n+16)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 9 + 10*x + x^2, P(3,x) = 20 + 57*x + 22*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 9 + 10*x + x^2;
P[3, x_] := 20 + 57*x + 22*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*P[n - 4, x] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*P[n - 3, x] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*P[n - 2, x] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*P[n - 1, x])/(4*n^2 + 4*n + 8)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 9 + 10*x + x^2; v[4] = 20 + 57*x + 22*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*v[i - 4] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*v[i - 3] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*v[i - 2] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*v[i - 1])/(4*n^2 + 4*n + 8)); 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.