Real-rooted atlas

A173018 → reverse (shift +1) → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,x + n\right)\,P_{n}(x) = \left(4 - 18\,x + 32\,x^{2} - 22\,x^{3} + 4\,x^{4} - 2\,n + 11\,n x - 25\,n x^{2} + 16\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 8\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 10\,x^{2} + 16\,x^{3} - 10\,x^{4} + 2\,x^{5} - n x + 6\,n x^{2} - 8\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-8 + 21\,x - 14\,x^{2} + 2\,x^{3} + 3\,n - 11\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x + 5\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2213
356112
4123314160
527158144301360
65866112085306212520
71212502843971801791126120160

All 15 rows

Technical details
Node id
n-1ad8c898d89287ac · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d762c6db5469cb8f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:   1;
  2:   2,     1;
  3:   5,     6,      1;
  4:  12,    33,     14,      1;
  5:  27,   158,    144,     30,      1;
  6:  58,   661,   1208,    530,     62,      1;
  7: 121,  2502,   8439,   7180,   1791,    126,     1;
  8: 248,  8833,  51150,  78095,  37084,   5775,   254,   1;
  9: 503, 29718, 279310, 712994, 597360, 175882, 18122, 510, 1;
%F Formula
The row polynomials satisfy (n-3+2*x)*P(n,x) = -(2*n-4+(2*n^2-11*n+18)*x+(n^3-8*n^2+25*n-32)*x^2+(3*n^2-16*n+22)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((n-2)*x+(n^2-6*n+10)*x^2-(n^2-8*n+16)*x^3-(3*n-10)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-8+(2*n^2-11*n+21)*x+(5*n-14)*x^2+2*x^3)*P(n-1,x) + ((n-3)*x-(n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (2*n^2 - 11*n + 18)*x + (n^3 - 8*n^2 + 25*n - 32)*x^2 + (3*n^2 - 16*n + 22)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((n - 2)*x + (n^2 - 6*n + 10)*x^2 - (n^2 - 8*n + 16)*x^3 - (3*n - 10)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 8 + (2*n^2 - 11*n + 21)*x + (5*n - 14)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 3)*x - (n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 3 + 2*x)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 4 + (2*n^2 - 11*n + 18)*x + (n^3 - 8*n^2 + 25*n - 32)*x^2 + (3*n^2 - 16*n + 22)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((n - 2)*x + (n^2 - 6*n + 10)*x^2 - (n^2 - 8*n + 16)*x^3 - (3*n - 10)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 8 + (2*n^2 - 11*n + 21)*x + (5*n - 14)*x^2 + 2*x^3)*v[i - 1] + ((n - 3)*x - (n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 3 + 2*x)); 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

Same fingerprint ?

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.