Real-rooted atlas

A111006 → D

nonnegative coefficients

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

Recurrence

\(\left(7\,x + 10\,x^{2} - n - 11\,n x - 15\,n x^{2} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-6 - 41\,x - 106\,x^{2} - 120\,x^{3} - 50\,x^{4} + 8\,n + 53\,n x + 133\,n x^{2} + 160\,n x^{3} + 75\,n x^{4} - 2\,n^{2} - 14\,n^{2} x - 39\,n^{2} x^{2} - 50\,n^{2} x^{3} - 25\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 29\,x^{2} - 70\,x^{3} - 70\,x^{4} - 25\,x^{5} + 4\,n x + 25\,n x^{2} + 61\,n x^{3} + 65\,n x^{4} + 25\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2145
304913
402152037
5009404089
60033610078217
700016110228147501

All 15 rows

Technical details
Node id
n-f84265f03f8449db · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-cb30a643bb214bd2

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: 1;
  2: 1, 4;
  3: 0, 4,  9;
  4: 0, 2, 15, 20;
  5: 0, 0,  9, 40,  40;
  6: 0, 0,  3, 36, 100,  78;
  7: 0, 0,  0, 16, 110, 228, 147;
  8: 0, 0,  0,  4,  70, 306, 497,  272;
  9: 0, 0,  0,  0,  25, 240, 777, 1040, 495;
%F Formula
The row polynomials satisfy (n^2-n+(4*n^2-11*n+7)*x+(5*n^2-15*n+10)*x^2)*P(n,x) = -(2*n^2-8*n+6+(14*n^2-53*n+41)*x+(39*n^2-133*n+106)*x^2+(50*n^2-160*n+120)*x^3+(25*n^2-75*n+50)*x^4)*P(n-1,x) + ((4*n-4)*x+(25*n-29)*x^2+(61*n-70)*x^3+(65*n-70)*x^4+(25*n-25)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 8*n + 6 + (14*n^2 - 53*n + 41)*x + (39*n^2 - 133*n + 106)*x^2 + (50*n^2 - 160*n + 120)*x^3 + (25*n^2 - 75*n + 50)*x^4)*P[n - 1, x] + ((4*n - 4)*x + (25*n - 29)*x^2 + (61*n - 70)*x^3 + (65*n - 70)*x^4 + (25*n - 25)*x^5)*D[P[n - 1, x], x])/(n^2 - n + (4*n^2 - 11*n + 7)*x + (5*n^2 - 15*n + 10)*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 8*n + 6 + (14*n^2 - 53*n + 41)*x + (39*n^2 - 133*n + 106)*x^2 + (50*n^2 - 160*n + 120)*x^3 + (25*n^2 - 75*n + 50)*x^4)*v[i - 1] + ((4*n - 4)*x + (25*n - 29)*x^2 + (61*n - 70)*x^3 + (65*n - 70)*x^4 + (25*n - 25)*x^5)*deriv(v[i - 1], x))/(n^2 - n + (4*n^2 - 11*n + 7)*x + (5*n^2 - 15*n + 10)*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.