Real-rooted atlas

A154233 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
289118
32057221100
440381288431753
5701591304410377415817
6112496928028182492984117154460
7168128371647862640858449273371741533880

All 15 rows

Technical details
Node id
n-67767e52c481da9f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d5053c86d2cadfa2

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:   8,     9,       1;
  3:  20,    57,      22,        1;
  4:  40,   381,     288,       43,        1;
  5:  70,  1591,    3044,     1037,       74,        1;
  6: 112,  4969,   28028,    18249,     2984,      117,       1;
  7: 168, 12837,  164786,   264085,    84492,     7337,     174,     1;
  8: 240, 29005,  709216,  3077519,  1869532,   317909,   16056,   247,   1;
  9: 330, 59299, 2458912, 23385013, 33862806, 10502655, 1016204, 32117, 338, 1;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^5-3*n^4-n^3+5*n^2+n-1)*x^2*P(n-4,x) - ((2*n^3-4*n-2)*x+(2*n^3-2*n^2-2*n)*x^2)*P(n-3,x) - (n+1-(2*n^3-6*n)*x+(n-1)*x^2)*P(n-2,x) + (2*n+(2*n-2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 8 + 9*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_] := 8 + 9*x + x^2;
P[3, x_] := 20 + 57*x + 22*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*P[n - 4, x] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*P[n - 3, x] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*P[n - 2, x] + (2*n + (2*n - 2)*x)*P[n - 1, x])/(n - 1)]];
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] = 8 + 9*x + x^2; v[4] = 20 + 57*x + 22*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*v[i - 4] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*v[i - 3] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*v[i - 2] + (2*n + (2*n - 2)*x)*v[i - 1])/(n - 1)); 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.