Real-rooted atlas

A052553 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(5 + 5\,x - 4\,x^{2} - 5\,n - 9\,n x + 4\,n x^{2} - n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(15 + 17\,x - 16\,x^{2} - 16\,x^{3} - 12\,n - 31\,n x - 6\,n x^{2} + 8\,n x^{3} - 4\,n^{2} + 8\,n^{2} x + 8\,n^{2} x^{2} + 3\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-15\,x - 47\,x^{2} - 48\,x^{3} - 16\,x^{4} - 10\,n - 38\,n x - 44\,n x^{2} - 16\,n x^{3} - 5\,n^{2} - 9\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
111
211
3314
4527
5107118
61816337
7333812184
85982344179

All 15 rows

Technical details
Node id
n-d3749a6ea39db754 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1aa2d780018369b6

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   1;
   3:   3,   1;
   4:   5,   2;
   5:  10,   7,   1;
   6:  18,  16,   3;
   7:  33,  38,  12,  1;
   8:  59,  82,  34,  4;
   9: 105, 173,  93, 18, 1;
  10: 185, 352, 231, 60, 5;
%F Formula
The row polynomials satisfy (n^3-n^2-5*n+5+(4*n^2-9*n+5)*x+(4*n-4)*x^2)*P(n,x) = (3*n^3-4*n^2-12*n+15+(2*n^3+8*n^2-31*n+17)*x+(8*n^2-6*n-16)*x^2+(8*n-16)*x^3)*P(n-1,x) - (5*n^2+10*n+(9*n^2+38*n+15)*x+(4*n^2+44*n+47)*x^2+(16*n+48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 4*n^2 - 12*n + 15 + (2*n^3 + 8*n^2 - 31*n + 17)*x + (8*n^2 - 6*n - 16)*x^2 + (8*n - 16)*x^3)*P[n - 1, x] - (5*n^2 + 10*n + (9*n^2 + 38*n + 15)*x + (4*n^2 + 44*n + 47)*x^2 + (16*n + 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - n^2 - 5*n + 5 + (4*n^2 - 9*n + 5)*x + (4*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n^3 - 4*n^2 - 12*n + 15 + (2*n^3 + 8*n^2 - 31*n + 17)*x + (8*n^2 - 6*n - 16)*x^2 + (8*n - 16)*x^3)*v[i - 1] - (5*n^2 + 10*n + (9*n^2 + 38*n + 15)*x + (4*n^2 + 44*n + 47)*x^2 + (16*n + 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - n^2 - 5*n + 5 + (4*n^2 - 9*n + 5)*x + (4*n - 4)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.