Real-rooted atlas

type_b_eulerian → γ (scale 1/4) → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,x + n\right)\,P_{n}(x) = \left(-3 - 18\,x - 26\,x^{2} - 4\,x^{3} + 16\,x^{4} + n + 18\,n x + 32\,n x^{2} + 10\,n x^{3} - 8\,n x^{4} + 3\,n^{2} + 2\,n^{2} x - 7\,n^{2} x^{2} - 6\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 10\,x^{2} - 4\,x^{3} + 20\,x^{4} + 16\,x^{5} + 10\,n x^{2} + 22\,n x^{3} + 12\,n x^{4} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 12\,x - 18\,x^{2} + 8\,x^{3} - 2\,n + 4\,n x + 10\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x^{2} - 8\,x^{3} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2314
311516
45238595
530030261663
620402644845615590
7159602545211421138554218

All 15 rows

Technical details
Node id
n-2f49fdc8f2063e0e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-46ff41ee5acfd118

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:       1;
  2:       3,       1;
  3:      11,       5;
  4:      52,      38,       5;
  5:     300,     302,      61;
  6:    2040,    2644,     845,     61;
  7:   15960,   25452,   11421,   1385;
  8:  141120,  268128,  157368,  30108,  1385;
  9: 1391040, 3074400, 2248200, 610444, 50521;
%F Formula
The row polynomials satisfy (n+4*x)*P(n,x) = -(n^3-3*n^2-n+3+(2*n^3-2*n^2-18*n+18)*x+(n^3+7*n^2-32*n+26)*x^2+(6*n^2-10*n+4)*x^3+(8*n-16)*x^4)*P(n-2,x) + ((2*n^2-2)*x+(4*n^2+10*n-10)*x^2+(2*n^2+22*n-4)*x^3+(12*n+20)*x^4+16*x^5)*d/dx P(n-2,x) + (2*n^2-2*n-1+(2*n^2+4*n-12)*x+(10*n-18)*x^2+8*x^3)*P(n-1,x) - (2*n*x+(2*n+8)*x^2+8*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[(-(n^3 - 3*n^2 - n + 3 + (2*n^3 - 2*n^2 - 18*n + 18)*x + (n^3 + 7*n^2 - 32*n + 26)*x^2 + (6*n^2 - 10*n + 4)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] + ((2*n^2 - 2)*x + (4*n^2 + 10*n - 10)*x^2 + (2*n^2 + 22*n - 4)*x^3 + (12*n + 20)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (2*n^2 - 2*n - 1 + (2*n^2 + 4*n - 12)*x + (10*n - 18)*x^2 + 8*x^3)*P[n - 1, x] - (2*n*x + (2*n + 8)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(n + 4*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] = (-(n^3 - 3*n^2 - n + 3 + (2*n^3 - 2*n^2 - 18*n + 18)*x + (n^3 + 7*n^2 - 32*n + 26)*x^2 + (6*n^2 - 10*n + 4)*x^3 + (8*n - 16)*x^4)*v[i - 2] + ((2*n^2 - 2)*x + (4*n^2 + 10*n - 10)*x^2 + (2*n^2 + 22*n - 4)*x^3 + (12*n + 20)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (2*n^2 - 2*n - 1 + (2*n^2 + 4*n - 12)*x + (10*n - 18)*x^2 + 8*x^3)*v[i - 1] - (2*n*x + (2*n + 8)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(n + 4*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

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.